1

ITTC Symbols and Terminology List

Version 2021

June 2021

Supersedes all previous versions

Updated by the 29th ITTC Quality Systems Group

ITTC Symbols and Terminology List, Version 2021 Table of Contents:

1.3.3.2 Flow angles etc ...... 26 1. GENERAL ...... 5 1.3.3.3 Forces ...... 27 1.1 Fundamental Concepts ...... 5 1.3.3.4 Sectional coefficients ...... 27 1.1.1 Uncertainty ...... 5 1.3.4 Boundary Layers ...... 28 1.1.2 Coordinates and Space Related 1.3.4.1 Two-dimensional Boundary Layers Quantities ...... 9 ...... 28 1.1.2.1 Basic Quantities ...... 12 1.3.5 Cavitation ...... 29 1.1.3 Time and Frequency Domain 1.3.5.1 Flow parameters ...... 29 Quantities ...... 13 1.3.5.2 Flow fields ...... 29 1.1.3.1 Basic Quantities ...... 13 1.3.5.3 Pumps ...... 29 1.1.3.2 Complex Transforms ...... 14 1.4 Environmental Mechanics ...... 30 1.1.3.3 Complex Quantities ...... 14 1.4.1 Waves ...... 30 1.1.4 Random Quantities and Stochastic Processes ...... 15 1.4.1.1 Periodic waves ...... 30 1.1.4.1 Random Quantities ...... 15 1.4.1.2 Irregular waves ...... 30 1.1.4.2 Stochastic Processes ...... 15 1.4.1.3 Time Domain Analysis ...... 31 1.1.4.3 Probability Operators 1.4.1.4 Frequency Domain Analysis ...... 31 (Superscripts) ...... 16 1.4.1.5 Directional Waves ...... 32 1.1.5 Balances and System Related 1.4.2 Wind ...... 33 Concepts ...... 17 1.4.2.1 Basic Quantities ...... 33 1.2 Solid Body Mechanics ...... 18 1.4.3 Ice Mechanics ...... 34 1.2.1 Inertial and Hydrodynamic 1.4.3.1 Basic Quantities ...... 34 Properties ...... 18 1.5 Noise ...... 35 1.2.1.1 Basic Quantities ...... 18 1.5.1 Underwater Noise ...... 35 1.2.2 Loads ...... 19 1.2.2.1 External Loads ...... 19 2. SHIPS IN GENERAL ...... 36 1.2.2.2 Sectional Loads ...... 20 2.1 Basic Quantities ...... 36 1.2.3 Rigid Body Motions ...... 21 2.2 Geometry and Hydrostatics ...... 38 1.2.3.1 Motions ...... 21 2.2.1 Hull Geometry ...... 38 1.2.3.2 Attitudes ...... 22 2.2.1.1 Basic Quantities ...... 38 1.3 Fluid Mechanics ...... 23 2.2.1.2 Derived Quantities ...... 39 1.3.1 Flow Parameters ...... 23 2.2.1.3 Symbols for Attributes and 1.3.1.1 Fluid Properties ...... 23 Subscripts ...... 40 1.3.1.2 Flow parameters ...... 23 2.2.2 Propulsor Geometry ...... 42 1.3.1.3 Boundary conditions...... 24 2.2.2.1 Screw Propellers ...... 42 1.3.2 Flow Fields ...... 25 2.2.2.2 Ducts ...... 44 1.3.2.1 Velocities etc...... 25 2.2.2.3 Waterjets (see also section 1.3.5)44 1.3.2.2 Circulation etc...... 25 2.2.2.4 Pods ...... 44 1.3.3 Lifting Surfaces ...... 26 2.2.2.5 Operators and identifiers ...... 44 1.3.3.1 Geometry ...... 26 2.2.3 Appendage Geometry ...... 45

3 2.2.3.1 Basic Quantities ...... 45 2.4.3 Large Amplitude Motions 2.2.3.2 Identifiers for Appendages Capsizing ...... 70 (Subscripts) ...... 45 2.4.4 Symbols for Attributes and 2.2.4 Hydrostatics and Stability ...... 47 Subscripts ...... 75 2.2.4.1 Points and Centres (Still under 3. SPECIAL CRAFT ...... 76 construction) ...... 47 2.2.4.2 Static Stability levers ...... 47 3.1 Planing and Semi-Displacement Vessels ...... 76 2.2.4.3 Derived Quantities ...... 49 3.1.1 Geometry and Hydrostatics ...... 76 2.2.4.4 Intact and Damage (Flooded) Stability ...... 49 3.1.2 Geometry and Levers, Underway ...... 77 2.2.4.5 Symbols for Attributes and Subscripts (under construction) .. 50 3.1.2.1 Geometry, Underway ...... 77 3.1.2.2 Levers, Underway ...... 78 2.3 Resistance and Propulsion ...... 51 3.1.3 Resistance and Propulsion ...... 79 2.3.1 Hull Resistance (see also Section 1.4.1 on Waves) ...... 51 3.2 Multi-Hull Vessels ...... 80 2.3.1.1 Basic Quantities ...... 51 3.2.1 Geometry and Hydrostatics ...... 80 2.3.1.2 Derived Quantities ...... 52 3.2.2 Resistance and Propulsion ...... 81 2.3.1.3 Symbols for Attributes and 3.2.2.1 Resistance Components ...... 81 Subscripts ...... 53 3.3 Hydrofoil Boats ...... 82 2.3.2 Ship Performance ...... 54 3.3.1 Geometry and Hydrostatics ...... 82 2.3.2.1 Basic Quantities ...... 54 3.3.1.1 Geometry, Underway ...... 82 2.3.2.2 Derived Quantities ...... 54 3.3.2 Resistance and Propulsion ...... 84 2.3.2.3 Efficiencies etc...... 55 3.3.2.1 Basic Quantities ...... 84 2.3.3 Propulsor Performance ...... 57 3.3.2.2 Derived Quantities ...... 85 2.3.3.1 Basic Quantities ...... 57 3.4 ACV and SES ...... 86 2.3.3.2 Derived Quantities ...... 57 3.4.1 Geometry and Hydrostatics ...... 86 2.3.3.3 Induced Velocities etc...... 59 3.4.2 Resistance and Propulsion ...... 87 2.3.4 Unsteady Propeller Forces ...... 60 3.5 Ice Going Vessels ...... 88 2.3.4.1 Basic Quantities ...... 60 3.5.1 Resistance and Propulsion ...... 88 2.3.5 Water Jets ...... 61 3.6 Sailing Vessels ...... 89 2.4 Manoeuvrability and Seakeeping ... 64 3.6.1 Geometry and Hydrostatics ...... 89 2.4.1 Manoeuvrability ...... 64 3.6.2 Resistance and Propulsion ...... 90 2.4.1.1 Geometrical Quantities ...... 64 2.4.1.2 Motions and Attitudes ...... 64 4. BACKGROUND AND REFERENCES 2.4.1.3 Flow Angles etc...... 65 ...... 91 2.4.1.4 Forces and Derivatives ...... 65 4.1 Symbols and Terminology Group .. 91

2.4.1.5 Linear Models ...... 66 4.2 Description of the List of Symbols . 91 2.4.1.6 Turning Circles ...... 66 4.2.1 Classification ...... 91 2.4.1.7 Zig-Zag Manoeuvres ...... 67 4.2.2 Structure of the Lists ...... 91 2.4.1.8 Stopping Manoeuvres ...... 67 4.2.3 Organization of the matrix 2.4.2 Seakeeping ...... 68 structured list ...... 91 2.4.2.1 Basic Quantities ...... 68 5. PRINCIPLES OF NOTATION ...... 92

4 5.1 Excerpts of ISO 31 ...... 93 5.3.2 Translations ...... 103 5.2 Computer Symbols ...... 102 5.3.3 Other References ...... 103

5.3 Documentation ...... 103 5.3.1 ITTC Documents ...... 103

NOTE: bold letters are used to denote vectors Red colour identifies the additions/modifications of this ver4sion of the List

ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.1 Uncertainty 5

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1. GENERAL 1.1 Fundamental Concepts 1.1.1 Uncertainty (The following table follows ISO/IEC Guide 98-3:2008 – Annex J) Half-width of a rectangular dis- Half-width of a rectangular dis- tribution of possible values of in- a tribution put quantity Xi: a = (a+– a_) / 2 Upper bound, or upper limit, of a+ Upper bound input quantity Xi: Lower bound, or lower limit, of a_ Lower bound input quantity Xi: Upper bound, or upper limit, of b+ Upper bound of the deviation the deviation of input quantity Xi from its estimate xi: b+ = a+ - xi Lower bound, or lower limit, of b_ Lower bound of the deviation the deviation of input quantity Xi from its estimate xi: b_ = xi – a_

ci Sensitivity coefficient ci  f/xi . 1 Functional relationship between measurand Y and input quantities X on which Y depends, and be- f Function i 1 tween output estimate y and in- put estimates xi on which y de- pends. Partial derivative of f with re- f/xi Partial derivative 1 spect to input quantity xi For calculation of expanded un- k Coverage factor 1 certainty U = kuc(y) For calculation of expanded un- kp Coverage factor for probability p 1 certainty Up = kpuc(y) n Number of repeated observations 1 Number of input quantities X on N Number of input quantities i 1 which the measurand Y depends p Probability Level of confidence: 0  p  1.0 1 q Random quantity 1 q Arithmetic mean or average 1 kth independent repeated obser- qk kth observation of q vation of randomly varying 1 quantity q r(xi,xj) Estimated correlation coefficient r(xi, xj) = u(xi, xj)/(u(xi) u(xj)) 1 2 sp Pooled estimate of variance 1 Pooled experimental standard de- s Positive square root of 2 1 p viation sp s2 (q)  s2 (q )/n ; estimated Experimental variance of the k s 2 (q ) 1 mean variance obtained from a Type A evaluation Experimental standard deviation s(q) Positive square root of s 2 (q ) 1 of the mean 2 Experimental variance from re- s (q ) 1 k peated observations ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.1 Uncertainty 6

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Experimental standard deviation 2 s(q ) Positive square root of s q 1 k of repeated observations k

From mean X i , determined 2 Experimental variance of input from n independent repeated ob- s (X ) 1 i mean servations Xi,k, estimated vari- ance obtained from a Type A evaluation. 2 s(X i ) Standard deviation of input mean Positive square root of s (X 1) 1 s(q,r ) Estimate of covariance of means 1 Estimate of covariance of input s(X i ,X ) 1 j means Student t-distribution for v de- tp(v) Inverse Student t grees of freedom corresponding 1 to a given probability p Student t-distribution for veff de- grees of freedom corresponding Inverse Student t for effective de- t (v ) to a given probability p in calcu- 1 p eff grees of freedom lation of expanded uncertainty Up

2 Associated with input estimate xi u (xi) Estimated variance 1 that estimates input quantity Xi 2 u(xi) Standard deviation Positive square root of u (xi) 1 u(xi,xj) Estimated covariance 1 2 Combined variance associated u (y) Combined variance 1 c with output estimate y u (y) Combined standard uncertainty 2 1 c Positive square root of uc (y) Combined standard uncertainty u (y) From Type A evaluations alone 1 cA from Type A Combined standard uncertainty u (y) From Type B evaluations alone 1 cB from Type B Combined standard uncertainty of output estimate yi when two uc(yi) Combined standard uncertainty or more measurands or output 1 quantities are determined in the same measurement 2 Component of combined vari- 2 2 u (y) u (y)  [c u(x )] 1 1 i ance i i i Component of combined stand- u (y) u (y)  c u(x ) 1 1 i ard uncertainty i i i Relative standard uncertainty of u(x )/|x | 1 i i output estimate x Relative combined standard un- u (y)/|y| c certainty of output estimate y

2 Estimated relative variance asso- [u(xi)/|xi|] Estimated relative variance ciated with input estimate xi Relative combined variance as- [u (y)/|y|]2 Relative combined variance c sociated with output estimate y Estimated relative covariance as- u(xi ,xj))/|xi xj| Estimated relative covariance sociated with input estimates xi and xj ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.1 Uncertainty 7

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Expanded uncertainty of output estimate y that defines an inter- val Y = y ±U having a high level U Expanded uncertainty of confidence, equal to coverage factor k times the combined standard uncertainty uc(y) of y: U = k uc(y) Expanded uncertainty of output estimate y that defines an inter- val Y = y ±U having a high level Expanded uncertainty associated p U of confidence p, equal to cover- p to confidence level p age factor kp times the combined standard uncertainty uc(y) of y: Up = kp uc(y) Estimate of input quantity Xi NOTE when xi is determined from the arithmetic mean or av- erage of n independent repeated

observation xii X xi Estimate of input quantity Xi

ith input quantity on which meas-

th urand Y depends Xi i input quantity NOTE Xi may be the physical quantity or the random variable Estimate of the value of input quantity X equal to the arithme- Estimate of the value of input i tic mean or average of n inde- quantity Xi pendent repeated observation Xi,k of Xi kth independent repeated observa- Xi,k tion of Xi Estimated of measurand Y or Re- y sult of a measurement or Output estimate Estimate of measurand Yi when two or more measurands are de- y Estimate of measurand Y i i termined in the same measure- ment A measurand. Estimated relative Y uncertainty of standard uncer- tainty u(xi) of inputs estimate xi Expectation or mean of the prob- Expectation or mean of the prob- µ ability distribution of random- p ability distribution varying quantity q ν Degrees of freedom (general) Degrees of freedom, or effective ν Degrees of freedom i degrees of freedom of standard ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.1 Uncertainty 8

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

uncertainty u(xi) of input esti- mate xi Effective degrees of freedom of u (y) used to obtain t (ν ) for ν Effective degrees of freedom c p eff eff calculating expanded uncertainty Up Variance of a probability distri- bution of (for example) a ran- 2 Variance of a probability domly-variing quantity q, esti- 2 mated by s (qk) Standard deviation of a probabil- Standard deviation of a probabil-  ity distribution, equal to the pos- ity distribution itive square root of 2 Variance of q , equal to  2 / n, estimated by 2 2 sq k   q Variance of q sq2   n

Standard deviation of , equal  q Standard deviation of to the positive root of Variance of experimental stand-  2 s q  ard deviation sq  of q Standard deviation of experi- mental standard deviation  s q of q , equal to the positive  square root of  2 s q    

ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.2 Coordinates and Space related Quantities 9

Coordinate systems fixed in bodies, ocean platforms, 1.1.2 Coordinates and Space Related Quanti- or ships. ties For the definition of hull forms and ocean wave prop- Orientation of coordinates erties and the analysis of structural deflections it is cus- tomary to take the x-axis positive forward and parallel to A problem of general interest, the orientation of the the reference or base line used to describe the body's axes of coordinate systems, has been treated extensively shape, the y-axis positive to port, and the z-axis positive in the Report of the 17th ITTC Information Committee. upwards. The present QS Group recommends that the orientations of the coordinate systems chosen for convenience should For seakeeping and manoeuvrability problems the co- be stated explicitly in any case. The coordinate system ori- ordinate system is defined as follows: usually the x-axis entation should not be inferred from the symbols and/or as before the y-axis positive to starboard, and the z-axis names of the concepts or from national or professional tra- positive downwards, the origin customarily at the centre ditions. All sign conventions of related Quantities should of mass of the vehicle or at a geometrically defined posi- be consistent with the orientation chosen. tion.

For ready reference the recommendation of the 17th For ship flow calculations usually the x-axis positive ITTC Information Committee is quoted in the following. in the main flow direction, i.e. backwards, the y-axis pos- itive to starboard, and the z-axis positive upwards, the "In order to adapt ITTC nomenclature to common origin customarily at the intersection of the plane of the practice a proposal for a standard coordinate system was undisturbed free-surface, the centre plane, and the mid- published in the newsletter No 7, March 1983, to generate ship section. discussion. The response was quite diverse. On the one hand it was suggested that instead of the two orthogonal right handed systems with the positive x-axis forward and Fixed or space axes (x0,y0,z0) the positive z-axis either up- or downward as proposed only one system should be selected, in particular the one Coordinate systems fixed in relation to the earth or with the positive z-axis upwards. On the other hand the the water. For further references see ISO Standard 1151/1 attention of the Information Committee was drawn to the ...6: Terms and symbols for flight dynamics. fact that in ship flow calculations neither of the two sys- tems proposed is customary. Normally the x-axis is di- rected in the main flow direction, i.e. backwards, the y- axis is taken positive to starboard and the z-axis is positive There may be other coordinate systems in use and upwards. The origin of the co-ordinates in this case is usu- there is no possibility for the adoption of a single system ally in the undisturbed free surface half way between fore for all purposes. Any problem requires an adequate coor- and aft perpendicular. dinate system and transformations between systems are simple, provided that orientations and origins are com- In view of this state of affairs the Information Com- pletely and correctly documented for any particular case." mittee (now Quality System Group - QSG) may offer the following recommendation, if any: Origins of coordinates

Axes, coordinates In seakeeping and manoeuvrability problems custom- arily the centre of mass of the vehicle is chosen as the Preferably, orthogonal right handed systems of Car- origin of the coordinates. This is in most cases not neces- tesian co-ordinates should be used, orientation and origin sarily advantageous, as all the hydrodynamic properties in any particular case should be chosen for convenience. entering the problems are related rather to the geometries of the bodies under investigation. So any geometrically Body axes (x,y,z) defined point may be more adequate for the purposes at hand.

ISO Standard 31-11 makes the following suggestions

ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.2 Coordinates and Space related Quantities 10

Item No. Coordinates Position vector and its differ- Name of coordi- Remarks ential nate system 11-12.1 x, y, z r = xex + yey + zez cartesian ex, ey and ez form an (-) dr = dx ex + dy ey + dz ez coordinates orthonormal right-handed sys- tem. See Figure 1. 11-12.2 ρ, φ, z r= ρeρ + zez cylindrical eρ(φ), eφ(φ) and ez form an or- (-) dr = dρ eρ + dφ eφ + dz ez coordinates thonormal right-handed system. See Figures 2 and 3. If z = 0, then ρ and φ are the polar coor- dinates 11-12.3 r=rer; spherical r,  , φ er(9, gyp), e ( , φ) and eφ(φ) (-) coordinates dr = dr er + r d e + form an orthonormal right-

r sin  dφ eφ handed system. See Figures 2 and 4. NOTE 1 If, exceptionally, a left-handed system is used for certain purposes, this shall be clearly stated to avoid the risk of sign errors.

ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.2 Coordinates and Space related Quantities 11

Figure 3 – Right-handed cylindrical coordinates

Figure 1 – Right-handed Cartesian coordinate system

Figure 4 – Right-handed spherical coordinates

Figure 2 – Oxyz is a right-handed coordinate system

ITTC Symbols 1 General 1.1 Fundamental Concepts Version 2021 1.1.2 Coordinates and Space related Quantities 12

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.1.2.1 Basic Quantities

Any scalar quantity distributed, s S ds maybe singularly, in space th 0 Zero order moment of a scalar S ij SM0(I,J) δ ds =δ S quantity ij ij First order moment of a scalar 1 S ij SM1(I,J) quantity, formerly static mo- εikjxkds ments of a scalar distribution Second moment of a scalar 2 S ij SM2(I,J) quantity, formerly moments of εklixlεjkm xmds inertia of a scalar distribution 0 Sij = S ij 1 T Si, 3+j = S ij Generalized moment of a scalar 1 Suv S(U,V) S3+i, j = S ij quantity distributed in space 2 S3+i, 3+j = S ij

Tensor in space referred to an s a Tij T(I,J) orthogonal system of Cartesian Tij + Tij coordinates fixed in the body A Tij TAS(I,J) Anti-symmetric part of a tensor ( Tij - Tji ) / 2 S Tij TSY(I,J) Symmetric part of a tensor ( Tij + Tji ) / 2 T Tij TTR(I,J) Transposed tensor Tji Tij vj Tensor product  Tij vj ui, vi U(I), V(I) Any vector quantities ui vi UVPS Scalar product uivi ui vj UVPD(I,J) Diadic product uivj uv UVPV(I) Vector product εijkujvk Zeroth order moments of a vec- tor quantity distributed in space, 0 V i ,Vi V0(I),V(I) referred to an orthogonal system dvi of Cartesian coordinates fixed in the body

1 First order moments of a vector V i V1(I) ε x dv distribution ijk j k 0 Vi = V i Vu V(U) Generalized vector 1 V3+i = V i x, x X, X(1) 1 Body axes and corresponding Right-hand orthogonal system of y, x Y, X(2) m 2 Cartesian coordinates coordinates fixed in the body z, x3 Z, X(3) x , x X0, X0(1) Right-hand orthogonal system of 0 01 Space axes and corresponding y , x Y0, X0(2) coordinates fixed in relation to m 0 02 Cartesian coordinates z0, x03 Z0, X0(3) the space x , x XF, XF(1) Right-hand orthogonal system of F F1 Flow axes and corresponding y , x YF, XF(2) coordinates fixed in relation to m F F2 Cartesian coordinates zF, xF3 ZF, XF(3) the flow +1 : ijk = 123, 231, 312 εijk EPS(I,J,K) Epsilon operator 1 : ijk = 321, 213, 132 0 : if otherwise +1 : ij = 11, 22, 33 δ DEL(I,J) Delta operator ij 0 : if otherwise ITTC Symbols 1 Mechanics in General 1.1 Fundamental Concepts Version 2021 1.1.3 Time and Frequency Domain Quantities 13

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.1.3 Time and Frequency Domain Quantities 1.1.3.1 Basic Quantities a ADMP Damping sr, in Laplace variable 1/s f FR Frequency Hz fC FC Basic frequency in repeating 1 / TC Hz functions fS FS Frequency of sampling 1 / TS Hz period in repeating spectra i I Imaginary unit sqrt(-1) 1 I IM Imaginary variable 1 j J Integer values - ...+ 1 R R Complex variable exp(s TS) Laurent transform s S Complex variable a + 2πif 1/s Laplace transform t TI Time - ... + s tj TI(J) Sample time instances j TS TC TC Period of cycle 1 / fC duration of cycles in peri- s odic, repeating processes TS TS Period of sampling Duration between samples s x x Values of real quantities x(t) X Real "valued" function

xj X(J) Variables for samples values of x(tj) = x(t)δ(t - tj)dt real quantities z Z Complex variable

ITTC Symbols 1 Mechanics in General 1.1 Fundamental Concepts Version 2021 1.1.3 Time and Frequency Domain Quantities 14

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.1.3.2 Complex Transforms xA XA Analytic function XA(t) = X(t) + iXH(t) DF DF x XDF Fourier transform of sampled X (f) = xjexp(-i2πfjTS) function i.e. periodically repeating F = X(0)/2 + fSX (f + jfS) sample theorem: aliasing! DL DL x XDL Laurent transform of X (s) = xjexp(-sjTS) sampled function xF XFT Fourier transform XF(f) = X(t)exp(-i2πft)dt inverse form: = XF(f)exp(-i2πft)dt if X(t) = 0 and a = 0 then XF(f)=XL(f) F x j XFT(J) Fourier transform of 1/TCX(t)exp(-i2πjt/TC)dt periodic function t = 0 . . TC F F X = x jδ(f - j/TC) inverse form: F X(t) = x jexp(-i2πfjTC) xH XHT Hilbert transform XH(t) = 1/π X(τ)/(t - τ)dτ xHF XHF Fourier transform of XHF(f) = XF(f)(-i sgn f) Hilbert transform (1/t)F = -i sgn f xL XLT Laplace transform XL(s) = X(t)exp(-st)dt if X(t<0) = 0 then = (X(t)exp(-at))F R R -j DL x XRT Laurent transform X (r) = xjr =X xS XS Single-sided complex spectra XS(f) = XF(f)(1 + sgn f) = XAF i.e. = 0 for f < 0 S F x j XS(J) Single-sided complex Fourier se- X j(1 + sgn j) ries line spectra

1.1.3.3 Complex Quantities za ZAM Amplitude mod(z) = sqrt(zr2+zi2) zc ZRE Real or cosine component zc = real(z) = zacos(zp) zi ZIM Imaginary or sine component imag(z) = zasin(zp) = zs zj ZCJ Conjugate zr - izi zl ZLG (Phase) Lag zp ZPH Phase arc(z) = arctg(zi / zr) zr ZRE Real or cosine component real(z) = zacos(zp) = zc zs ZIM Imaginary or sine component zs = imag(z) = zasin(zp) ITTC Symbols 1 Mechanics in General 1.1 Fundamental Concepts Version 2021 1.1.4 Stochastic Processes 15

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.1.4 Random Quantities and Stochastic Processes 1.1.4.1 Random Quantities E M MR g , g , g GMR Expected value of a function of a E(g) = g(x)fx(x)dx random quantity x = - ...  x, y X, Y Random quantities x(ζ), y(ζ) xi, yi X(I), Y(I) Samples of random quantities i = 1... n n : sample size (xm)E XmMR m-th moment of a random quan- (xm)E tity D DR VR ½ x , x , σx XDR Standard deviation of a random x quantity DS VS 1/2 x , sx XDS Sample deviation of a random x , quantity unbiased random estimate of the standard deviation R MR E xx , xx , Rxx XXMR Auto-correlation of a random x x quantity R MR E xy , xy , Rxy XYMR Cross-correlation of two random x y quantities E M MR x , x , x , μx XMR Expectation or population mean E(x) of a random quantity A MS x , x , mx XMS Average or sample mean of a 1/n  xi , i = 1...n random quantity unbiased random estimate of the expectation with xAE = xE xVSE = xV / n PD x , fx XPD Probability density of a random d Fx / dx quantity PD 2 xy , fxy XYPD Joint probability density of two  Fxy / (x y) random quantities PF x , Fx XPF Probability function (distribu- 1 tion) of a random quantity PF xy , Fxy XYPF Joint probability function (distri- 1 bution) function of two random quantities xV, xVR, xxVR XVR, XXVR Variance of a random quantity x2 E - xE 2 VS VS A 2 x , xx XVS, XXVS Sample variance of a random 1/ (n - 1)  (xi - x ) quantity i = 1...n unbiased random estimate of the variance xVSE = xV xyV, xyVR XYVR Variance of two random quanti- x yE - xE yE ties ζ Outcome of a random "experi- ment"

1.1.4.2 Stochastic Processes gMR GMR Mean of a function of a random M(g(t)) = lim(1/T g(t)dt) quantity t =-T/2 ... +T/2 T =- ... + gMS GMS Average or sample mean of a A(g(t)) = 1/T g(t)dt function of a random quantity t =0 ... +T x, y X, Y Stationary stochastic process x(ζ,t), y(ζ,t) C CR E E E xx , xx , Cxx XXCR Auto-covariance of a stationary (x(t) - x )(x(t + τ) - x ) stochastic process ITTC Symbols 1 Mechanics in General 1.1 Fundamental Concepts Version 2021 1.1.4 Stochastic Processes 16

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

C CR E E E xy , xy , Cxy XYCR Cross-covariance of two station- (x(t) - x )(y(t + τ) - y ) ary stochastic processes R RR E xx , xx , Rxx XXRR Auto-correlation of a stationary x(t)x(t + τ) = Rxx(τ) stochastic process Rxx(τ) = Rxx(-τ) if x is ergodic: MR Rxx(τ) = x(t)x(t + τ) Rxx(τ) =  Sxx(ω)cos(ωτ)dτ τ = 0 ...  R E xy , Rxy XYRR Cross-correlation of two station- x(t)y(t + τ) = Rxy(τ) ary stochastic processes Ryx(τ) = Rxy(-τ) if x, y are ergodic: MR Rxy(τ) = x(t)y(t + τ) S RRSR xx , Sxx XXSR Power spectrum or autospectral xx power density of a stochastic process S RRSR xy , Sxy XYSR Cross-power spectrum of two xy stationary stochastic processes τ TICV Covariance or correlation time s ζ Outcome of a random "experi- ment"

1.1.4.3 Probability Operators (Superscripts) A, MS MS Average, sample mean C, CR CR Population covariance CS CS Sample covariance D, DR DR Population deviation DS DS Sample deviation E, M, MR MR Expectation, population mean PD PD Probability density PF PF Probability function S SR (Power) Spectrum SS SS Sample spectrum R, RR RR Population correlation RS RS Sample correlation V, VR VR Population variance VS VS Sample variance ITTC Symbols 1 Mechanics in General 1.1 Fundamental Concepts Version 2021 1.1.5 Balances and System Related Concepts 17

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.1.5 Balances and System Related Concepts

Quantity of the quality under q QQ consideration stored in a control QU volume Q Quality under consideration QU/s QC QCF Convective flux QU/s QD QDF Diffusive flux QU/s Total flux across the QF QFL Inward positive! QU/s surface of the control volume QM Molecular diffusion QU/s Production of sources in the con- QP QPN QU/s trol volume Storage in the control volume, QS QRT rate of change of the quantity dq / dt QU/s stored QT QDT Turbulent diffusion QU/s

ITTC Symbols 1 Mechanics in General 1.2 Solid Body Mechanics Version 2021 1.2.1 Inertial and Hydrodynamic Properties 18

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.2 Solid Body Mechanics 1.2.1 Inertial and Hydrodynamic Properties 1.2.1.1 Basic Quantities

Added mass coefficient in ith A AM(I,J) ij mode due to jth motion Damping coefficient in ith mode B DA(I,J) ij due to jth motion Restoring force coefficient in ith C RF(I,J) ij mode due to jth motion

h Generalized hydrodynamic h D uv DH(U,V) F / V damping uv h F u FH(U) Generalized hydrodynamic force

h Generalized hydrodynamic iner- h I uv IH(U,V) F /V  tia u v Longitudinal second moment of About transverse axis through I IL m4 L water-plane area centre of floatation Transverse second moment of About longitudinal axis through I IT m4 T water-plane area centre of floatation IY, IYY, I , I , y yy M2(2,2), Pitch moment of inertia m2 , kg m2 22 MA(5,5) around the principal axis y m 55 I , I , IZ, IZZ, z zz Yaw moment of inertia m2 , M2(3,3), kg m2 33 around the principal axis z m66 MA(6,6) I , I IXY, I2(1,2) xy 12 Real products of inertia in case I , I IYZ, I2(2,3) kg m2 yz 23 of non-principal axes Izx , I31 IZX, I2(3,1) k , k Roll radius of gyration around x xx RDGX (I /m)1/2 m k the principal axis x xx Pitch radius of gyration k , k RDGY (I /m)1/2 m y yy around the principal axis y yy Yaw radius of gyration around k , k RDGZ (I /m)1/2 m z zz the principal axis z zz m MA mass kg M0(I,J), m0 , Zero-th moments of mass, i.e. in- ij MA(I,J) m = m δ kg m ertia distribution, mass tensor ij ij ij First moments of mass, i.e. iner- m1 M1(I,J) Alias static moments of mass kg m ij tia distribution m2 , M2(I,J), Second moments of mass, i.e. in- ij Alias mass moments of inertia kg m2 Iij IN(I,J) ertia distribution 0 Generalized mass, i. e. general- Mij = M ij 1T ized inertia tensor of a (rigid) Mi, 3+j = M ij Muv MA(U,V) 1 body referred to a body fixed co- M3+i, j = M ij 2 ordinate system M3+i , 3+j = M ij ITTC Symbols 1 Mechanics in General 1.2 Solid Body Mechanics Version 2021 1.2.2 Loads 19

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.2.2 Loads 1.2.2.1 External Loads

F M M u = M u Force, generalized, load, 0 Fu F(U) Fi = F i N in body coordinates 1 F3+i = F i 1 Gravity field strength, general- gi = g i 2 gu G(U) m/s ized, in body coordinates g3+i = 0 Gravity field strength, g G1(I) m/s2 i in body coordinates! K, Mx , K, M(1), 1 Moment around body axis x Nm F 1 , F4 F1(1), F(4) M, My , M, M(2), 1 Moment around body axis y Nm F 2 , F5 F1(2), F(5) N, Mz , N, M(3), 1 Moment around body axis z Nm FN 3 , F6 F1(3), F(6) X, Fx , X, FX, 0 Force in direction of body axis x Nm F 1 , F1 F0(1), F(1) Y, Fy , Y, FY, 0 Force in direction of body axis y Nm F 2 , F2 F0(2), F(2) Z, Fz , Z, FZ, 0 Force in direction of body axis z Nm F 3 , F3 F0(3), F(3) Gravity or weight force, general- G G(U) G = m g N u ized, in body co-ordinates! u uv v 0 0 0 Gravity or weight force Gi = G i = m ij gj G i , Gi G0(I) N in body coordinates! = mgi 1 0 1 Gravity or weight moment in G3+i = G i = εikj xk G j G i G1(I) 1 Nm body coordinates! = m ij gj q UNQ Load per unit length N/m w WPUL Weight per unit length dW / dx1 N/m ITTC Symbols 1 Mechanics in General 1.2 Solid Body Mechanics Version 2021 1.2.2 Loads 20

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.2.2.2 Sectional Loads

Force or load acting at a given S S0 S planar cross-section of the body, F i = F i N F u FS(U) S S1 B generalized, in section coordi- F 3+i = F i = M i Nm nates! S S0 S0 F i FS(I) Shearing force F 2 , F 3 N FT, FT Tensioning or normal force FS0 N FS(1) 1 B S1 S1 M i MB(I) Bending moment F 2 , F 3 Nm MT, MT Twisting or torsional moment FS1 Nm MB(1) 1

ITTC Symbols 1 Mechanics in General 1.2 Solid Body Mechanics Version 2021 1.2.3 Rigid Body Motions 21

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.2.3 Rigid Body Motions 1.2.3.1 Motions L L Angular momentum L = I (= r2mv) Kg m s-1 P P Linear momentum P = m v Kg m s-1 p , ωx , P, OMX, Rotational velocity around body 0 rad/s v 1 , v4 V0(1), V(4) axis x q , ωy , Q, OMY, Rotational velocity around body 0 rad/s v 2 , v5 V0(2), V(5) axis y r , ωz , R, OMZ, Rotational velocity around body 0 rad/s v 3 , v6 V0(3), V(6) axis z u , vx , U, VX, Translatory velocity in the direc- 1 m/s v 1 , v1 V1(1), V(1) tion of body axis x v , vy , V, VY, Translatory velocity in the direc- 1 m/s v 2 , v2 V1(2), V(2) tion of body axis y w , vz , W, VZ, Translatory velocity in the direc- 1 m/s v 3 , v3 V1(3), V(3) tion of body axis z Components of generalized ve- v = v1 m/s v V(U) locity or motion relative to body i i u v = v0 rad/s axes 3+i i p PR Rates of change of components q QR of rotational velocity relative to rad/s2 RR body axes r u UR Rates of change of components v VR of linear velocity relative to m/s2 w WR body axes α AA Angular acceleration dω/dt rad/s2

ITTC Symbols 1 Mechanics in General 1.2 Solid Body Mechanics Version 2021 1.2.3 Rigid Body Motions 22

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.2.3.2 Attitudes

The angle of the longitudinal body axis from the projection into the principal plane of sym- AT metry of the velocity of the α Angle of attack rad ALFA origin of the body axes relative to the fluid, positive in the posi- tive sense of rotation about the y-axis The angle to the principal plane of symmetry from the velocity DR vector of the origin of the body β Angle of drift or side-slip rad BET axes relative to the fluid, posi- tive in the positive sense of rota- tion about the z-axis The angular displacement about the x axis of the principal plane RO o γ Projected angle of roll or heel of symmetry from the vertical, rad GAMR positive in the positive sense of rotation about the xo axis X(4) Positive in the positive sense of φ RO Angle of roll, heel or list rad rotation about the x-axis PHIR X(5) Positive in the positive sense of θ TR Angle of pitch or trim rad rotation about the y-axis TETP X(6) Positive in the positive sense of ψ YA Angle of yaw, heading or course rad rotation about the z-axis PSIY

ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2021 1.3.1 Flow Parameters 23

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3 Fluid Mechanics 1.3.1 Flow Parameters 1.3.1.1 Fluid Properties c CS Velocity of sound (E / ρ)1/2 m/s E EL Modulus of elasticity Pa w WD Weight density ρg (See 1.1.1) κ CK Kinematic capillarity σ / ρ m3/s2 μ VI Viscosity kg/ms v VK Kinematic viscosity μ / ρ m2/s ρ DN, RHO Mass density kg/m3 σ CA Capillarity Surface tension per unit length kg/s2

1.3.1.2 Flow parameters 1/2 Bo BN Boussinesq number V / (g RH) 1 Ca CN Cauchy number V / (E / ρ)1/2 1 Fr FN Froude number V / (g L)1/2 1 1/2 Frh FH Froude depth number V / (g h) 1 1/3 1/2 Fr FV Froude displacement number V / (g  ) 1 Ma MN Mach number V / c 1 Re RN Reynolds number V L / v 1 2 Propeller Reynolds number at 2 Re RN07 cV0.7 A  0.7 nD1 0.7 0.7 R Re  0.7  St SN Strouhal number f L / V 1 Thoma number, Cavitation num- Th TN (p – p )/q 1 ber A V We WN Weber number V2 L / κ 1

ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2021 1.3.1 Flow Parameters 24

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3.1.3 Boundary conditions Roughness height, usually in k HK Roughness height or magnitude m terms of some average Mean diameter of the equivalent k SK Sand roughness m s sand grains covering a surface Area of section divided by wet- R RH Hydraulic radius m H ted perimeter ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2021 1.3.2 Flow Fields 25

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3.2 Flow Fields 1.3.2.1 Velocities etc. e ED Density of total flow energy ρ V2 / 2 + p + ρ g h Pa

Strength of force fields, usually 2 fi FS(I) Mass specific force m/s only gravity field gi Δz , h HS Static pressure head 0 m z0-axis positive vertical up! H HT Total head e / w = h +p/w +q/w m Pressure, density of static flow p PR, ES Pa energy Ambient pressure in undisturbed p P0 Pa 0 flow Dynamic pressure, density of ki- q PD, EK ρ V2 / 2 Pa netic flow energy, QF, Volume passing across a control Q Rate of flow m3/s QFLOW surface in time unit SH THL Total head loss m R CR s ij SR(I,J) Turbulent or Reynolds stress ρ vivj Pa Density of total diffusive mo- sij ST(I,J) Total stress tensor mentum flux due to molecular Pa and turbulent exchange V s ij SV(I,J) Viscous stress Pa u, v ,v VX, V1 x 1 Velocity component in direction v, v ,v VY, V2 m/s y 2 of x, y, z axes w, vz ,v3 VZ, V3 vi V(I) Velocity m/s 1/2 V VA Velocity V = vivi m/s V0 V0 Velocity of undisturbed flow m/s τw TAUW Wall shear stress μ (U / y)y=0 Pa 1.3.2.2 Circulation etc. Γ / (π D V) Γn CN Normalized circulation 1 π is frequently omitted Ratio between velocities induced I ID Induction factor by helicoidal and by straight line 1 vortices Strength per length or per area of Γ VD Vortex density m/s vortex distribution ∫V ds Γ CC Circulation m2/s along a closed line Φ PO Potential function m2/s ψ = const Ψ SF Stream function is the equation of a stream sur- m3/s face

ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2021 1.3.3 Lifting Surfaces 26

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3.3 Lifting Surfaces 1.3.3.1 Geometry 2 A AP Projected area b cM m b SP Wing or foil span m bF BSPF Flap span m cM CHME Mean chord length A / b m cT CHTP Tip chord length m cr CHRT Root chord length m fL FML Camber of lower side (general) m fU FMU Camber of upper side m γ ANSW Sweep angle rad δs ANSL Slat deflection angle rad Thickness ratio of foil section δ DELTT t / c 1 (general) Thickness ratio of trailing edge δ DELTB t / c 1 B of struts B S Camber ratio of mean line (gen- δ DELTF f / c 1 F eral) δFL DLTFL Angle of flap deflection rad Camber ratio of lower side of δ DELTL f / c 1 L foil L δS DELTS Thickness ratio of strut tS / cS 1 Theoretical thickness ratio of δ DELTT t / c 1 STH section S STH δU DELTU Camber ratio of upper side fU / c 1 λ TA Taper ratio ct / cr 1 Λ AS Aspect ratio b2 / A 1 1.3.3.2 Flow angles etc VI VI Induced velocity m/s Resultant velocity of flow ap- Taking vortex induced velocities V VT m/s T proaching a hydrofoil into account Angle between the direction of AA, α Angle of attack or incidence undisturbed relative flow and the rad ALFA chord line The angle of attack relative to AAEF, Effective angle of attack or inci- α the chord line including the ef- rad EFF ALFE dence fect of induced velocities The angle of attack relative to AAGE, Geometric angle of attack or in- α the chord line neglecting the ef- rad G ALFG cidence fect of induced velocities AAHY, In relation to the position at zero α Hydrodynamic angle of attack rad H ALFI lift For thin airfoil or hydrofoil, an- gle of attack for which the streamlines are tangent to the AAID, α Ideal angle of attack mean line at the leading edge. rad I ALFS This condition is usually referred to as "shock-free" entry or "smooth" AAZL, Angle of attack or incidence at α Angle of zero lift rad 0 ALF0 zero lift

ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2021 1.3.3 Lifting Surfaces 27

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3.3.3 Forces Force in the direction of motion D DRF Foil drag N F of an immersed foil For finite span foil, the compo- DI DRIND Induced drag nent of lift in the direction of N motion Due to mutual interaction of the DINT DRINT Interference drag boundary layers of intersecting N foil Section or profile drag at zero D DRSE Streamline drag N P lift LF LF Lift force on foil CL AFT q N Lift force for angle of attack of L LF0 C A q N 0 zero L0 FT 1.3.3.4 Sectional coefficients CD CDSE Section drag coefficient 1 CDI CDSI Section induced drag coefficient 1 CL CLSE Section lift coefficient 1 Section lift coefficient for angle C CLSE0 1 L0 of attack of zero CM CMSE Section moment coefficient 1 ε EPSLD Lift-Drag ratio L/D 1

ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2014 1.3.4 Boundary Layers 28

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3.4 Boundary Layers 1.3.4.1 Two-dimensional Boundary Layers 2 Cf CFL Skin friction coefficient τ / (ρ Ue / 2) 1 F CQF Entrainment factor 1 / (Ue dQ / dx) 1 H HBL Boundary layer shape parameter δ* / Θ 1 * HE HQF Entrainment shape parameter (δ - δ ) / Θ 1 p PR Static pressure Pa P PT Total pressure Pa b Q QF Entrainment Udy m2/s a

Reynolds number based on dis- * * Reδ* RDELS U δ / v or U δ / v 1 placement thickness e Reynolds number based on mo- Reθ RTHETA U Θ / v or U Θ / v 1 mentum thickness e Velocity fluctuations in bound- u UFL m/s ary layer Root mean square value of ve- us UFLS m/s locity fluctuations Non-dimensional distance from u+ UPLUS U / u 1 surface τ 1/2 uτ UTAU Shear (friction) velocity (τ / ρ) m/s Time mean of velocity in bound- U UMR m/s m ary layer Ui UIN Instantaneous velocity m/s Free-stream velocity far from the U UFS m/s surface Velocity at the edge of the Ue UE m/s boundary layer at y=δ995 Velocity defect in boundary ΔU UDEF (U - U) / u 1 layer e τ Non-dimensional distance from y+ YPLUS y u / v 1 the wall τ * β BETE Equilibrium parameter δ / (τw dp / dx) 1 Thickness of a boundary layer at δ995 DEL m U=0.995Ue

* Displacement thickness of δ , δ1 DELS (U - U) / U dy m boundary layer e e K K von Karman constant 0.41 1 Λ PRGR Pressure gradient parameter δ995 / (v dUe / dx) 1 * ** 2 2 θ , δ ENTH Energy thickness (U / Ue) (1 - U / Ue )dy m Θ THETA Momentum thickness (U / Ue) (1 - U / Ue)dy m τw TAUW Local skin friction μ (U / y)y=0 Pa

ITTC Symbols 1 Mechanics in General 1.3 Fluid Mechanics Version 2021 1.3.5 Cavitation 29

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.3.5 Cavitation 1.3.5.1 Flow parameters as GR Gas content ratio α / αS 1 Actual amount of solved and un- α GC Gas content ppm dissolved gas in a liquid Maximum amount of gas solved α GS Gas content of saturated liquid ppm S in a liquid at a given temperature σ CNPC Cavitation number (pA - pC) / q 1 σI CNPI Inception cavitation number 1 σV CNPV Vapour cavitation number (pA - pV) / q 1 1.3.5.2 Flow fields DC DC Cavity drag N Stream wise dimension of a lC LC Cavity length fully-developed cavitating re- m gion pA PA Ambient pressure Pa Absolute ambient pressure at p PACO Collapse pressure Pa AC which cavities collapse Absolute ambient pressure at pAI PAIC Critical pressure which cavitation inception takes Pa place Pressure within a steady or p PC Cavity pressure Pa C quasi-steady cavity Pressure, may be negative, i.e. pCI PCIN Initial cavity pressure tensile strength, necessary to cre- Pa ate a cavity pV PV Vapour pressure of water At a given temperature! Pa Free stream velocity at which U UNIN Critical velocity m/s I cavitation inception takes place 3 VL VOLS Volume loss WL / w m Weight of material eroded from WL WTLS Weight loss a specimen during a specified N/s time Maximum height of a fully-de- veloped cavity, normal to the δ HC Cavity height or thickness m C surface and the stream-wise di- rection of the cavity 1.3.5.3 Pumps HN HTNT Net useful head of turbo-engines m Total head upstream of turbo-en- H HTUS m U gines Th,  TN Thoma number (HU - pV / w) / HN 1 ITTC Symbols 1 Mechanics in General 1.4 Environmental Mechanics Version 2021 1.4.1 Waves 30

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.4 Environmental Mechanics 1.4.1 Waves This section is related to Sections 3.1.2 Time and Frequency Domain Quantities and 3.1.3 Random Quantities and Sto- chastic Processes. 1.4.1.1 Periodic waves

gLW /2 cW VP Wave phase velocity or celerity LW / TW = in m/s deep water const = c Wave phase velocity of harmonic W c VP(I) for periodic waves in deep m/s Wi components of a periodic wave water The average :rate of ad- vance of the energy in a c VG Wave group velocity or celerity m/s G finite train of gravity waves fW FW Basic wave frequency 1 / TW Hz Frequencies of harmonic compo- f FW(I) i f Hz Wi nents of a periodic wave W The vertical distance from wave crest to wave trough, or H HW Wave height m W twice the wave amplitude of a harmonic wave. ηC - ηT 2 k, κ WN Wave number 2 π / LW = ω /g 1/m The horizontal distance be- LW , λW LW Wave length tween adjacent wave crests in m the direction of advance Time between the passage of TW TW Basic wave period two successive wave crests s past a fixed point. 1 / fW The angle between the direc- µ WD Wave direction tion of a component wave and rad the x0 axis Instantaneous wave elevation at a z-axis positive vertical up, η EW m given location zero at mean water level; Amplitudes of harmonic compo- ηα EWAM(I) ηFSα m i nents of a periodic wave Phases of harmonic components ηp , ε EWPH(I) ηFSp rad i i of a periodic wave ηC EC Wave crest elevation m ηT ET Wave trough depression Negative values! m z-axis positive vertical down, ζ DW Instantaneous wave depression m zero at mean water level Radius of orbital motion of a ζ WAMP Wave amplitude m A surface wave particle ωW, σ FC Circular wave frequency 2 π fW = 2 π / TW rad/s

1.4.1.2 Irregular waves Wave height by zero down- The vertical distance between a H HD m d crossing successive crest and trough. The vertical distance between a H HU Wave height by zero up-crossing m u successive trough and crest Average of the highest one third H H13 Significant wave height m 1/3 wave heights ITTC Symbols 1 Mechanics in General 1.4 Environmental Mechanics Version 2021 1.4.1 Waves 31

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

T1/3d T13D Significant wave period By downcrossing analysis s T1/3u T13U Significant wave period By upcrossing analysis s Time elapsing between two suc- Wave periods by zero down- T TD cessive downward crossings of s d crossing zero in a record Time elapsing between two suc- Wave periods by zero up-cross- T TU cessive upward crossings of zero s u ing in a record Maximum of elevations of wave η EC m C crests in a record Elevations of wave troughs in a η ET Negative values! m T record The horizontal distance between Wave length by zero down- λ LD adjacent down crossing in the di- m d crossing rection of advance The horizontal distance between λu LU Wave length by zero up-crossing adjacent up crossing in the direc- m tion of advance 1.4.1.3 Time Domain Analysis Wave height estimated from vis- H HWV m WV ual observation Significant wave height. Sum of Average of the highest one third H H13 significant wave height of swell m W1/3 wave heights and wind waves Average of the highest one third H H13S Significant wave height of swell m 1/3S wave heights of the swell. (environmental mechanics, Average of the highest one third H H13W waves) Significant wave height m 1/3W wave heights of the wind waves. of wind waves. Zero up-crossing significant Average of the highest one third H H13U m 1/3u wave height zero up-crossing wave heights Estimate of significant wave Hσ HWDS height from sample deviation of m wave elevation record Wave length estimated by visual Measured in the direction of L LWV m WV observation wave propagation The average interval in years be- Trt TRT Return period tween times that a given design wave is exceeded TR TR Duration of record 1 / fR s 1 / fS , TS TS Sample interval time between two successive s samples Wave period estimated from vis- T TWV s WV ual observation 1.4.1.4 Frequency Domain Analysis

Sampling frequency divided by b B Bandwidth of spectral resolution Hz the number of transform points Cr CRA Average reflection coefficient 1 Reflection coefficient amplitude C (f) CRF 1 r function Frequency at which the spectrum f FRPK Spectral peak in frequency Hz P has its maximum ITTC Symbols 1 Mechanics in General 1.4 Environmental Mechanics Version 2021 1.4.1 Waves 32

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

fR FRRC Frequency resolution 1 / TR Hz fS FRSA Sample frequency 1 / TS Hz Significant wave height based on 1/2 Hmo HMO zeroth moment for narrow 4 (m0) m banded spectrum Estimate of significant wave Hσ HWDS height from sample deviation of m wave elevation record

n-th moment of wave power n 2 n mn MN f S(f)df m / s spectral density S (f), EISF, Incident wave power spectral i m2/Hz Si(ω) EISC density S (f), ERSF, Reflected wave power spectral r m2/Hz Sr(ω) ERSC density S (f), EWSF, η Wave power spectral density m2/Hz Sη(ω) EWSC TP TP Period with maximum energy 2πfP s Average period from zeroth and T T1 m /m s 01 first moment 0 1 Average period from zeroth and T T2 (m /m )1/2 s 02 second moment 0 2 1.4.1.5 Directional Waves S(f,θ)=S(f)DX(f,θ) where DX(f,θ), DIRSF 2 DX(ω,μ), Directional spreading function rad DX f , d  1 σθ SIGMAOX  0 f FR Frequency 2πω=1/T Hz S (ω,μ) S2ZET ζ Two dimensional spectral den- S (ω,μ) S2TET 1 θ sity etc. etc. S (f,θ) m2/Hz/ ρ STHETA Directional spectral density Sζ(ω,μ) rad θ, μ CWD Component wave direction rad MWD Mean or dominant wave direc- θ rad m THETAMOX tion

ITTC Symbols 1 Mechanics in General 1.4 Environmental Mechanics Version 2021 1.4.2 Wind 33

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.4.2 Wind 1.4.2.1 Basic Quantities -3 C10 C10M Surface drag coefficient (0.08 + 0.065U10)10 Distance over water the wind F FETCH Fetch length m blows td DURATN Wind duration s The average interval in years be- Trt TRT Return period tween times that a given wind speed is exceeded uz , uzi UFLUCT Turbulent wind fluctuations m/s 1/2 1.23 UA, u* USHEAR Wind shear velocity C10 U10 or 0.71U10 m/s Reference mean wind speed at 1/7 A U10 U10M elevation 10 meters above sea U10 = (10/z) Uz m/s surface A (Uz + uzi) A Average wind speed at elevation A 1/7 Uz UZA Uz = (z/10) U10 or m/s z above the sea surface A Uz = U10 + UA ln(z/10) VWR VWREL Apparent wind velocity see section 1.4.1 m/s VWT VWABS True wind velocity see section 1.4.1 m/s 2 XF FDIM Dimensionless Fetch gF/U10 Height above the sea surface in z ZSURF m meters Apparent wind angle (relative to β BETWA see section 2.6 rad WA vessel course) True wind angle (relative to ves- β BETWT see section 2.6 rad WT sel course) θW TETWI Wind direction rad ITTC Symbols 1 Mechanics in General 1.4 Environmental Mechanics Version 201 1.4.3 Ice Mechanics 34

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

1.4.3 Ice Mechanics 1.4.3.1 Basic Quantities EI MEI Modulus of elasticity of ice Pa Weight of salt per unit weight of S SAIC Salinity of ice 1 I ice Weight of dissolved salt per unit S SAWA Salinity of water 1 W weight of saline water tA TEAI Temperature of air C tI TEIC Local temperature of ice C tW TEWA Temperature of water C δI ELIC Deflection of ice sheet Vertical elevation of ice surface m εI STIC Ice strain Elongation per unit length 1

 I STRTIC Ice strain rate ε / τ 1/s μI POIIC Poisson's ratio of ice 1 Volume of gas pores per unit vol- v POAI Relative volume of air 1 A ume of ice Volume of liquid phase per unit v POBR Relative volume of brine 1 B volume of ice v0 POIC Total porosity of ice v0 = vA + vB 1 3 ρI DNIC Mass density of ice Mass of ice per unit volume kg/m 3 ρSN DNSN Mass density of snow Mass of snow per unit volume kg/m 3 ρW DNWA Mass density of water kg/m 3 ρΔ DNWI Density difference ρΔ = ρW - ρI kg/m σCI SCIC Compressive strength of ice Pa σFI SFIC Flexural strength of ice Pa σTI SNIC Tensile strength of ice Pa τSI STIC Shear strength of ice Pa ITTC Symbols 1 Mechanics in General 1.5 Noise Version 2021 1.5.1 Underwater noise 35

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit  1.5 Noise 1.5.1 Underwater Noise Distance hydrophone to acoustic 𝑑 DIDR m centre 𝑝̅ 𝐿 10log dB, 𝑝 𝐿 SPL Sound pressure level 𝑝 1Pa 𝐿s Underwater sound radiated noise 𝐿p 𝐿 SRNL s level at a reference distance of 1m 𝑑 20log dB,𝑑 1 m 𝑑 p SPRE Sound pressure Pa

ITTC Symbols 2 Ships in General

Version 2021 2.1 Basic Quantities 36

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit  2. SHIPS IN GENERAL 2.1 Basic Quantities a, a1 AC, A1 Linear or translatory acceleration dv / dt m/s2 A, AR, A Area in general m2 AREA B B, BR Breadth m F C, F 2 FF(2) Cross force Force normal to lift and drag (forces) N C Cc CC Cross force coefficient C  1 C qA Force opposing translatory velocity, F D, F 1 FF(1) Resistance, Drag (force) generally for a completely immersed N body d, D D, DI Diameter m E E, EN Energy J f FR Frequency 1 / T Hz F, F0 F, F0 Force N Weight force / mass, strength of the g G, GR Acceleration of gravity m/s2 earth gravity field h DE Depth m H H, HT Height m Second order moment of a mass dis- I I, IN Moment of inertia kg m2 tribution L L, LE Length m Force perpendicular to translatory ve- L, FF FF(3) Lift (force) N 3 locity M, MA, m Mass kg MASS First order moment of a force distri- M, F1 M1, F1 Moment of forces Nm bution M MO Momentum Ns Alias RPS (RPM in some propulsor n, N FR, N Frequency or rate of revolution Hz applications) P P, PO Power W r, R RD Radius m F R, F 1 R, RE, FF(1) Resistance (force) Force opposing translatory velocity N s SP Distance along path m t TI Time s t TE Temperature K Duration of a cycle of a repeating or T TC Period periodic, not necessarily harmonic s process U U, UN Undisturbed velocity of a fluid m/s Linear or translatory velocity of a v, V1 V, V1 ds / dt m/s body V VO Volume m3 Weight density, formerly specific w WD dW / dV = ρg N/m3 weight Weight (force), gravity force act- W WT N ing on a body Relative mass or weight, in Eng- Mass density of a substance divided γ MR lish speaking countries called spe- by mass density of distilled water at 1 cific gravity 4C η EF, ETA Efficiency Ratio of powers ITTC Symbols 2 Ships in General

Version 2021 2.1 Basic Quantities 37

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit  ρ DN, RHO Mass density dm / dV kg/m3 water density for reference water  RHO0 kg/m3 0 temperature and salt content 3 ρA DNA, RHOA Mass density of air Mass of air per unit volume kg/m τ ST, TAU Tangential stress Pa Ship dimension divided by corre- λ SC Scale ratio 1 sponding model dimension σ SN, SIGS Normal stress Pa ω FC, OMF Circular frequency 2 π f 1/s ω, V0 V0, OMN Rotational velocity 2 π n rad/s ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.1 Hull Geometry 38

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.2 Geometry and Hydrostatics 2.2.1 Hull Geometry 2.2.1.1 Basic Quantities The area of the ram projected on Area of bulbous bow in longitudi- A ABL the middle line plane forward of m2 BL nal plane the fore perpendicular The cross sectional area at the fore perpendicular. Where the water lines are rounded so as to termi- Area of transverse cross-section of nate on the forward perpendicular A ABT a bulbous bow (full area port and m2 BT A is measured by continuing the star-board) BT area curve forward to the perpen- dicular, ignoring the final round- ing. Midway between fore and aft per- A AM Area of midship section m2 M pendiculars Area of transom (full area port and Cross-sectional area of transom A ATR m2 T starboard) stern below the load waterline Area of portion of ship above wa- 2 AV AV Area exposed to wind terline projected normally to the m direction of relative wind Transverse projected area above Projected area of the ship above the 2 𝐴XV AXV the waterline including superstruc- waterline projected on a transversal m tures plane 2 AW AW Area of water-plane m 2 AWA AWA Area of water-plane aft of midship m Area of water-plane forward of A AWF m2 WF midship Area of maximum transverse sec- A AX m2 X tion B B Breadth, moulded, of ships hull m Breadth, moulded of midship sec- B BM m M tion at design water line Breadth, moulded of transom at B BTR m T design water line Maximum moulded breadth at de- B BWL m WL sign water line Breadth, moulded of maximum B BX m X section area at design water line d, T T Draught, moulded, of ship hull m T - T alias “keel drag” d KDROP Design drop of the keel line AD FD m KL or “slope of keel” D DEP Depth, moulded, of a ship hull m From the freeboard markings to f FREB Freeboard the freeboard deck, according to m official rules Angle of waterline at the bow with iE ANEN Angle of entrance, half reference to centre plane, neglect- rad ing local shape at stem Angle of waterline at the stern with reference to the centre-plane, i ANRU Angle of run, half rad R neglecting local shape of stern frame ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.1 Hull Geometry 39

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Reference length of ship (gener- L L Length of ship ally length between the perpendic- m ulars) From the forward perpendicular to LE LEN Length of entrance the forward end of parallel middle m body, or maximum section LOA LOA Length, overall m LOS LOS Length, overall submerged m Length of constant transverse sec- L LP Length of parallel middle body m P tion LPP LPP Length between perpendiculars m From section of maximum area or after end of parallel middle body L LRU Length of run m R to waterline termination or other designated point of the stern LWL LWL Length of waterline m LFS LFS Frame spacing used for structures m LSS LSS Station spacing m S S, AWS Area of wetted surface m2 The intercept of the tangent to the t TT Taylor tangent of the area curve sectional area curve at the bow on 1 the midship ordinate T, d T Draught, moulded, of ship hull m TA, dA TA, TAP Draught at aft perpendicular m Design draught at aft perpendicu- T TAD, TAPD m AD lar TF, dF TF, TFP Draught at forward perpendicular m Design draught at forward perpen- T TFD, TFPD m FD dicular Maximum draught of the hull T THUL Draught of the hull m H without keel or skeg (T + T ) / 2 for rigid bodies with T , d TM, TMS Draught at midship A F m M M straight keel TMD TMD, TMSD Design draught at midship (TAD + TFD) / 2 for rigid bodies m Vertical depth of trailing edge of TT TTR Immersion of transom boat at keel below water surface m level 3 , V DISPVOL Displacement volume Δ / (ρ g) = BH + AP m 3 BH DISPVBH Displacement volume of bare hull ΔBH / (ρ g) m

Displacement volume of append- 3  DISPVAP ΔAP / (ρ g) m APP ages Δ DISPF Displacement force (buoyancy) g ρ  N Displacement force (buoyancy) of ΔBH DISPFBH g ρ  N bare hull BH Displacement force (buoyancy) of ΔAPP DISPFAP g ρ  N appendages AP Δm DISPM Displacement mass ρ kg λ = L / L = B / B λ SC Linear scale of ship model S M S M 1 = TS / TM 2.2.1.2 Derived Quantities BC CIRCB R.E. Froude's breadth coefficient B / 1/3 1 CB CB Block coefficient  / (L B T) 1

CGM CGM Dimensionless GM coefficient GM /1/3 1 ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.1 Hull Geometry 40

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

CGZ CGZ Dimensionless GZ coefficient GZ /1/3 1 CKG CKG Dimensionless KG coefficient KG /T 1 Coefficient of inertia of water C CWIL 12 I / ( B L3) 1 IL plane, longitudinal L Coefficient of inertia of water C CWIT 12 I / (B3 L) 1 IT plane, transverse T Midship section coefficient (mid- CM CMS way between forward and aft per- AM / (B T) 1 pendiculars)

CP CPL Longitudinal prismatic coefficient  / (AX L) or  / (AM L) 1 A / (AX L / 2) or CPA CPA Prismatic coefficient, after body 1 A / (AM L / 2) E / (AX LE) or CPE CPE Prismatic coefficient, entrance 1 E / (AM LE) F / (AX L / 2) or CPF CPF Prismatic coefficient fore body 1 F / (AM L / 2) R / (AX LR) or CPR CPR Prismatic coefficient, run 1 R / (AM LR) 1/2 CS CS Wetted surface coefficient S / (V L) 1 CVP CVP Prismatic coefficient vertical  / (AW T) 1 CWA CWA Water plane area coefficient, aft AWA / (B L / 2) 1 Water plane area coefficient, for- C CWF A /(B L / 2) 1 WF ward WF CWP CW Water plane area coefficient AW /(L B) 1 A / (B T), where B and T are Maximum transverse section coef- X C CX measured at the position of maxi- 1 X ficient mum area 3 C CVOL Volumetric coefficient  / L 1 fBL CABL Area coefficient for bulbous bow ABL / (L T) 1 Taylor sectional area coefficient f CABL A / A 1 BT for bulbous bow BT X Sectional area coefficient for tran- f ATR A / A 1 T som stern T X R.E. Froude's length coefficient, or MC CIRCM L / 1/3 1 length-displacement ratio R.E. Froude's wetted surface area SC CIRCS 23 1 coefficient S  TC CIRCT R.E. Froude's draught coefficient T / 1/3 1

2.2.1.3 Symbols for Attributes and Subscripts A AB After body AP AP After perpendicular APP APP Appendages B BH Bare hull DW Design waterline E EN Entry F FB Fore body FP FP Fore perpendicular FS FS Frame spacing H HE Hull LR Reference Line LP LP Based on LPP LW LW Based on LWL ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.1 Hull Geometry 41

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

M MS Midships PB Parallel body R RU Run SS SS Station spacing W WP Water plane S WS Wetted surface

ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.2 Propulsor Geometry 42

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.2.2 Propulsor Geometry 2.2.2.1 Screw Propellers Developed blade area of a screw A AD Developed blade area m2 D propeller outside the boss or hub Expanded blade area of a screw A AE Expanded blade area m2 E propeller outside the boss or hub 2 2 A0 AO Propeller Disc Area π D / 4 m Projected blade area of a screw A AP Projected blade area m2 P propeller outside the boss or hub aD ADR Developed blade area ratio AD / A0 1 aE ADE Expanded blade area ratio AE / A0 1 aP ADP Projected blade area ratio AP / A0 1 c LCH Chord length m c0.7 C07 Chord length Chord length at r/R=0.7 m The part of the Chord delimited by the Leading Edge and the intersec- cLE CHLE Chord, leading part tion between the Generator Line m and the pitch helix at the consid- ered radius The expanded or developed area cM CHME Mean chord length of a propeller blade divided by the m span from the hub to the tip The displacement between middle of chord and the blade reference cS CS Skew displacement line. Positive when middle chord m is at the trailing side regarding the blade reference line The part of the Chord delimited by the Trailing Edge and the intersec- cTE CHTE Chord, trailing part tion between the Generator Line m and the pitch helix at the consid- ered radius dh DH Boss or hub diameter 2 rh m Aft diameter of the hub, not con- d DHA Hub diameter, aft m ha sidering any shoulder Fore diameter of the hub, not con- d DHF Hub diameter, fore m hf sidering any shoulder D DP Propeller diameter m f FBP Camber of a foil section m GZ GAP Gap between the propeller blades 2 π r sin (φ) / z m The depth of submergence of the propeller measured vertically from h HO Immersion m 0 the propeller centre to the free sur- face Distance between the propeller H HTC Hull tip clearance m TC sweep circle and the hull The displacement from the propel- ler plane to the generator line in i , R (ISO) RAKG Rake m G k the direction of the shaft axis. Aft displacement is positive rake. The axial displacement of a blade section which occurs when the i RAKS Rake, skew-induced m S propeller is skewed. Aft displace- ment is positive rake ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.2 Propulsor Geometry 43

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

The axial displacement of the blade reference line from the pro- iT RAKT Rake, total peller plane m iG + iS = cS sinφ Positive direction is aft. The length of the hub, including l LH Hub length m h any fore and aft shoulder Length of the hub taken from the lha LHA Hub length, aft propeller plane to the aft end of m the hub including aft shoulder Length of the hub taken from the lhf LHF Hub length, fore propeller plane to the fore end of m the hub including fore shoulder NP NPR Number of propellers 1 p PDR Pitch ratio P / D 1 Ρ PITCH Propeller pitch in general m r LR Blade section radius m rh RH Hub radius m R RDP Propeller radius m t TM Blade section thickness m Thickness on axis of propeller Thickness of propeller blade as ex- t TO m 0 blade tended down to propeller axis xB XBDR Boss to diameter ratio dh / D Distance of propeller centre for- x XP Longitudinal propeller position m P ward of the after perpendicular Transverse distance of wing pro- y YP Lateral propeller position m P peller centre from middle line Z, z NPB Number of propeller blades 1 Height of propeller centre above z ZP Vertical propeller position m P base line Angle of inclination of the propel- Angle between propeller shaft and 𝛼 AA deg ler shaft horizontal Propeller axis angle measured to Angle between reference line and ε, ψbP PSIBP rad body fixed coordinates propeller shaft axis The angular displacement about the shaft axis of the reference point of any blade section relative θs TETS Skew angle to the generator line measured in rad the plane of rotation. It is positive when opposite to the direction of ahead rotation θ RAKA Angle of rake rad The difference between maximum θ TEMX Skew angle extent rad EXT and minimum local skew angle φ PHIP Pitch angle of screw propeller arctg (P / (2 π R)) 1 Pitch angle of screw propeller φ PHIF 1 F measured to the face line Propeller axis angle measured to Angle between horizontal plane ψaP PSIAP rad space fixed coordinates and propeller shaft axis τB Blade thickness ratio t0 / D 1

ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.2 Propulsor Geometry 44

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.2.2.2 Ducts 2 ADEN ADEN Duct entry area m 2 ADEX ADEX Duct exit area m Clearance between propeller tip d CLEARD Propeller tip clearance m D and inner surface of duct fD FD Camber of duct profile m LD LD Duct length m Axial distance between leading L LDEN Duct entry part length m DEN edge of duct and propeller plane Axial distance between propeller L LDEX Duct exit length m DEX plane and trailing edge of duct tD TD Thickness of duct profile m Angle between nose-tail line of α AD Duct profile-shaft axis angle rad D duct profile and propeller shaft Angle between inner duct tail line β BD Diffuser angle of duct rad D and propeller shaft 2.2.2.3 Waterjets (see also section 1.3.5) 2 An , A6 Nozzle discharge area m A Cross sectional area at station s m2 s D Impeller diameter (maximum) m D Nozzle discharge diameter m n H Head between station i and j m ij

PJSE H Jet System Head m JS QJ maximum height of cross sectional h m 1A area of stream tube at station 1A gH K Head coefficient: 1 H nD25 2.2.2.4 Pods Wetted Surface Area of Pod Main A APB m2 PB Body Wetted Surface Area of Bottom A APBF m2 PBF Fin 2 APS APS Wetted Surface Area of Strut m Thickness Cord Ratio of Bottom C CBFTC 1 BFTC Fin CSTC CSTC Thickness Cord Ratio of Strut 1 DPB DPB Maximum Diameter of Pod Body m LPB LPB Length of Pod Main Body m Code length of bottom fin under L LPBF Length of Bottom Fin m PBF pod main body Code length of strut between for- L LPS Length of Upper Strut m PS ward edge and aft edge TPBS TPBS Bottom Thickness of Strut m 2.2.2.5 Operators and identifiers a absolute (space) reference (superscript) b body axis reference (superscript) P propeller shaft axis (subscript) D Duct (subscript) ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.3 Appendage Geometry 45

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.2.3 Appendage Geometry Related information may be found in Section 3.3.3 on Lifting Surfaces. 2.2.3.1 Basic Quantities 2 AC AC Area under cut-up m 2 AFB AFB Area of bow fins m Projected frontal area of an ap- A AFR Frontal area m2 FR pendage 2 ARF AF Projected flap area m 2 AR ARU Lateral rudder area Area of the rudder, including flap m Lateral area of the fixed part of A ARX m2 RX rudder Lateral area of rudder in the pro- A ARP m2 RP peller race 2 ART ART Total lateral rudder area ARX + ARmov m 2 AFS AFS Projected area of stern fins m 2 ASK ASK Projected skeg area m 2 SWBK SWBK Wetted surface area of bilge keels m c CH Chord length of foil section m cM CHME Mean chord length ART / S m cR CHRT Chord length at the root m cT CHTP Chord length at the tip m Camber of an aerofoil or a hydro- Maximum separation of median f FM m foil and nose-tail line Measured in direction parallel to L LF Length of flap or wedge m F keel Maximum thickness of an aerofoil t TMX Measured normal to mean line m or a hydrofoil αFB ANFB Bow fin angle rad αFS ANFS Stern fin angle rad Angle between the planing surface δF DELFS Flap angle (general) of a flap and the bottom before the rad leading edge Angle between the planing surface δW DELWG Wedge angle of a wedge and the bottom before rad the leading edge δFR ANFR Flanking rudder angle rad Initial angle set up during the as- Assembly angle of flanking rud- δ ANFRIN sembly as zero angle of flanking rad FRin ders rudders δR ANRU Rudder angle rad δRF ANRF Rudder-flap angle rad λR TARU Rudder taper cT / cR 1 λFR TAFR Flanking rudder taper 1 2 ΛR ASRU Rudder aspect ratio bR / ART 1 ΛFR ASRF Flanking rudder aspect ratio 2.2.3.2 Identifiers for Appendages (Subscripts) BK Bilge keel BS Bossing FB Bow foil FR Flanking rudder FS Stern foil KL Keel RU Rudder RF Rudder flap ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.3 Appendage Geometry 46

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

SA Stabilizer SH Shafting SK Skeg ST Strut TH Thruster WG Wedge ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.4 Hydrostatics and Stability 47

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.2.4 Hydrostatics and Stability 2.2.4.1 Points and Centres (Still under construction) Assumed centre of gravity above A keel used for cross curves of sta- bility Centre of flotation of added buoy- b ant layer or centre of lost buoy- ancy of the flooded volume Centroid of the underwater vol- B Centre of buoyancy ume Centre of flotation of the water F plane Centre of gravity of an added or g removed weight (mass) G Centre of gravity of a vessel K Keel reference M Metacentre of a vessel See subscripts for qualification Longitudinal distance from refer- Longitudinal centre of buoyancy ence point to the centre of buoy- XCB , LCB XCB m (LCB) ancy, B such as XMCF from Mid- ships Longitudinal distance from refer- Longitudinal centre of flotation ence point to the centre of flota- XCF , LCF XCF m (LCF) tion, F such as XMCF from Mid- ships Longitudinal distance from refer- Longitudinal centre of buoyancy ence point to the centre of buoy- x XACB m Cb of added buoyant layer ancy of the added buoyant layer, b such as xMCb from Midships Longitudinal distance from refer- Longitudinal centre of flotation of ence point to the centre of flota- x XACF m Cf added buoyant layer tion of the added buoyant layer, f such as xMCf from Midships Longitudinal distance from refer- ence to the centre of gravity, g, of Longitudinal centre of gravity of x XACG an added or removed weight m Cg added weight (mass) (mass) such as xMCg from Mid- ships Longitudinal distance from a ref- Longitudinal centre of gravity erence point to the centre of grav- XCG , LCG XCG m (LCG) ity, G such as XMCG from Mid- ships Lateral displacement of centre of Lateral distance from a reference Y YCG m CG gravity (YCG) point to the centre of gravity, G Intersection of righting arm with Z ZRA line of action of the centre of buoyancy Vertical distance from reference 𝑍 ZCB Vertical centre of buoyancy m CB point to the centre of buoyancy, B

2.2.4.2 Static Stability levers Longitudinal centre of buoyancy Distance of centre of buoyancy XAB m AB from aft perpendicular from aft perpendicular ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.4 Hydrostatics and Stability 48

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Distance of centre of flotation XAF m AF from aft perpendicular Longitudinal centre of gravity Distance of centre of gravity from AG XAG m L from aft perpendicular aft perpendicular Transverse distance from assumed AG YAG centre of gravity A, to actual cen- m T tre of gravity G Vertical distance from assumed AG ZAG centre of gravity A, to actual cen- m V tre of gravity G Righting arm based on horizontal Generally tabulated in cross YAZ distance from assumed centre of m AZ curves of stability gravity A, to Z Distance from the centre of buoy- Transverse metacentre above cen- ancy B to the transverse metacen- ZBM m BM tre of buoyancy tre M. BM= I / = KM -KB T Longitudinal metacentre above ZBML BM L centre of buoyancy KM L - KB Longitudinal centre of buoyancy, Distance of centre of buoyancy XFB m FB LCB, from forward perpendicular from forward perpendicular Longitudinal centre of floatation, Distance of centre of flotation XFF m FF LCF, from forward perpendicular from forward perpendicular Longitudinal centre of gravity Distance of centre of gravity from XFG m FG from forward perpendicular forward perpendicular Horizontal stability lever caused GG H GGH by a weight shift or weight addi- m tion Longitudinal stability lever caused GG L GGL by a weight shift or weight addi- m tion Vertical stability lever caused by a GG1, GGV m GG1 , GG V weight shift or weight addition KG 1  KG 0  GG 1 Distance of centre of gravity to the GM Transverse metacentric height metacentre m GM KM - KG

Effective transverse metacentric GM corrected for free surface GM EFF GMEFF m height and/or free communication effects Distance from the centre of gravity Longitudinal centre of metacentric G to the longitudinal metacentre GML m GM L height ML

GM L  KM L  KG

GZ  AZ  AGV sin – GZ Righting arm or lever m GZ AG T cosφ

GZ MAX GZMAX Maximum righting arm or lever m Distance from the assumed centre Assumed centre of gravity above ZKA of gravity A to the moulded base m KA moulded base or keel or keel K ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.4 Hydrostatics and Stability 49

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Distance from the centre of buoy- Centre of buoyancy above ZKB ancy B to the moulded base or m KB moulded base or keel keel K Centre of gravity above moulded Distance from centre of gravity G ZKG m KG base or keel to the moulded base or keel K Vertical centre of gravity of added Distance from centre of gravity, g, Kg ZKAG or removed weight above moulded m to the moulded base or keel K base or keel Distance from the transverse meta- Transverse metacentre above ZKM centre M to the moulded base or m KM moulded base or keel keel K Distance from the longitudinal Longitudinal metacentre above ZKML metacentre M to the moulded m KM L moulded base or keel L base or keel K l XTA Longitudinal trimming arm xCG - xCB m t YHA Equivalent transverse heeling arm Heeling moment /Δ m 2.2.4.3 Derived Quantities

CGM CGM Dimensionless GM coefficient GM /1/3 1

CGZ CGZ Dimensionless GZ coefficient GZ /1/3 1 CKG CKG Dimensionless KG coefficient KG /T 1 Trimming moment divided by change in trim which approxi- CMTL CMTL Longitudinal trimming coefficient mately equals 1

BM L / L 2.2.4.4 Intact and Damage (Flooded) Stability trimming moment divided by change in trim which approxi- CMTL CMTL Longitudinal trimming coefficient mately equals 1

BML / L From the freeboard markings to f FREB Freeboard the freeboard deck, according to m official rules ASI, IAS ASI Attained subdivision index (to be clarified) 1 GZ Other moments such as Moment of ship stability in gen- those of capsizing, heeling, etc. M MS Nm S eral will be represented by MS with additional subscripts as appropri- ate m SHIPMA Ship mass W / g kg Moment to change trim by one M MTC Nm/cm TC centimetre Moment to change trim by one M MTM ΔC Nm/m TM meter MTL RSI RSI Required subdivision index 1 ts, tKL TRIM Static trim TA - TF - dKL m W SHIPWT Ship weight m g N zSF ZSF Static sinkage at FP Caused by loading m zSA ZSA Static sinkage at AP Caused by loading m zS ZS Mean static sinkage (zSF + zSA) / 2 m δ D Finite increment in... Prefix to other symbol 1 δtKL DTR Change in static trim m ITTC Symbols 2 Ships in General 2.2 Geometry and Hydrostatics Version 2021 2.2.4 Hydrostatics and Stability 50

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Δ DISPF Displacement (buoyant) force g ρ  N Δm DISPM Displacement mass ρ kg  DISPVOL Displacement volume Δ / (ρg) m3

Displacement volume of flooded 3  DISVOLFW Δfw / (ρg) m fw water -1 θS TRIMS Static trim angle tan ((zSF - zSA) / L) rad The ratio of the volume of flood- μ PMVO Volumetric permeability ing water in a compartment to the 1 total volume of the compartment  HEELANG Heel angle rad F HEELANGF Heel angle at flooding rad VS HEELANGV Heel angle for vanishing stability rad

2.2.4.5 Symbols for Attributes and Subscripts (under construction) a apparent A, att attained d, dyn dynamic e, EFF effective f false KL keel line L longitudinal MAX maximum MTL longitudinal trimming moment R, req required (to be clarified) s Static S, sqt Sinkage, squat TC Trim in cm TM Trim in m T transverse V vertical 0 Initial  at heel angle  θ at trim angle θ

ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.1 Hull Resistance 51

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.3 Resistance and Propulsion 2.3.1 Hull Resistance (see also Section 1.4.1 on Waves) 2.3.1.1 Basic Quantities Maximum transverse area of m BLCK Blockage parameter model ship divided by tank cross 1 section area Incremental resistance to be added to the smooth ship resistance to R RA Model-ship correlation allowance N A complete the model-ship predic- tion RAA RAA Air or wind resistance N RAPP RAP Appendage resistance N RAR RAR Roughness resistance N RTM[(1 + k) CFMC + CR] / Resistance corrected for difference [(1 + k) CFM + CR] RC RC in temperature between resistance where CFMC is the frictional coeffi- N and self-propulsion tests cient at the temperature of the self- propulsion test Due to fluid friction on the surface R RF Frictional resistance of a body N F of the body RF0 RF0 Frictional resistance of a flat plate N Due to the normal stresses over the R RP Pressure resistance N P surface of a body Due to normal stress related to vis- R RPV Viscous pressure resistance N PV cosity and turbulence RR RR Residuary resistance RT - RF or RT - RF0 N Residuary resistance of the bare R RRBH N RBH hull RS RS Spray resistance Due to generation of spray N RT RT Total resistance Total towed resistance N RTBH RTBH Total resistance of bare hull N RV RV Total viscous resistance RF + RPV N RW RW Wave making resistance Due to formation of surface waves N Associated with the breakdown of R RWB Wave breaking resistance N WB the bow wave RWP RWP Wave pattern resistance N 2 S S Wetted surface area, underway SBH + SAPP m 2 S0 S0 Wetted surface area, at rest SBH0 + SAPP0 m Appendage wetted surface area, S SAP m2 APP underway Appendage wetted surface area, at S SAP0 m2 APP0 rest Bare Hull wetted surface area, un- S SBH m2 BH derway Bare Hull wetted surface area, at S SBH0 m2 BH0 rest ΔCF DELCF Roughness allowance 1 V V Speed of the model or the ship m/s VK VKN Speed in knots VWR VWR Wind velocity, relative m/s zVF ZVF Running sinkage at FP m zVA ZVA Running sinkage at AP m ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.1 Hull Resistance 52

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

zVM ZVM Mean running sinkage (zVF + zVA) / 2 m η EW Wave Elevation see 3.4.1 m -1 θV , θD TRIMV Running (dynamic) trim angle tan ((zVF - zVA) / L) 1 2 τW LSF, TAUW Local skin friction see 3.3.4 N/ m 2.3.1.2 Derived Quantities Incremental resistance coefficient C CA 1 A for model ship correlation RAA / (S q)

CAA CAA Air or wind resistance coefficient A AV A AV 1 = CDA = CX S SS S SS

CAPP CAPP Appendage resistance coefficient RAPP / (S q) 1 CD CD Drag coefficient D / (S q) 1 Air or wind resistance coefficient, 𝑅AA CDA CDA 1 from wind tunnel tests 𝐴V A Frictional resistance coefficient of C CF R / (S q) 1 F a body F Frictional resistance coefficient of C CF0 R / (S q) 1 F0 a corresponding plate F0 Cp CP Local pressure coefficient 1 Pressure resistance coefficient, in- C CPR R / (S q) 1 PR cluding wave effect P Viscous pressure resistance coef- C CPV R / (S q) 1 PV ficient PV CR CR Residuary resistance coefficient RR / (S q) 1 CS CSR Spray resistance coefficient RS / (S q) 1 CT CT Total resistance coefficient RT / (S q) 1 2 CTL CTLT Telfer's resistance coefficient g R L / (ΔV ) 1 CTQ CTQ Qualified resistance coefficient CT / (ηH ηR) 1 2/3 CT CTVOL Resistance displacement RT / ( q) 1 Total viscous resistance coeffi- C CV R / (S q) 1 V cient V Wave making resistance coeffi- C CW R / (S q) 1 W cient W Wave pattern resistance coeffi- C CWP 1 WP cient, by wave analysis Air or wind resistance coefficient, C CXA -R / (A q ) 1 X usually from wind tunnel tests AA V R R.E. Froude's resistance coeffi- CC CIRCC 1000 R / (Δ(KC)2) 1 cient T R.E. Froude's frictional resistance FC CIRCF 1000 R / (Δ(KC)2) 1 coefficient F Ratio of tangential force to normal f FC Friction coefficient 1 force between two sliding bodies Three dimensional form factor on k K (C - C ) / C 1 flat plate friction V F0 F0 k(θ) WDC Wind direction coefficient CAA/ CAA0 1

C R.E. Froude's speed displacement 1/2 1/2 1/6 K CIRCK (4 π) Fr or (4π/g) V / coefficient K

Resistance coefficient corre- 4 2 KR KR R / (ρ D n ) 1 sponding to KQ, KT Dynamic pressure, density of ki- ρ V2 / 2 q PD, EK Pa netic flow energy, see 3.3.2 ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.1 Hull Resistance 53

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Dynamic pressure based on ap- ρ V 2/ 2 q PDWR, EKWR WR Pa R parent wind see 3.4.2 R. E. Froude’s wetted surface co- SC CIRCS 23 1 efficient S  Resistance-displacement ratio in ε EPSG R / Δ 1 general Residuary resistance-displace- ε EPSR R / Δ 1 R ment ratio R 2.3.1.3 Symbols for Attributes and Subscripts FW Fresh water MF Faired model data MR Raw model data OW Open water SF Faired full scale data SR Raw full scale data SW Salt water ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.2 Ship Performance 54

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.3.2 Ship Performance 2.3.2.1 Basic Quantities Transverse projected area above Projected area of the ship above the 2 𝐴XV AXV the waterline including superstruc- waterline projected on a transversal m tures plane Towing force applied to a model Friction deduction force in self to correct the model resistance for F SFC N D propulsion test different Re between model and full scale. FP FP Force pulling or towing a ship N FP0 FPO Pull during bollard test N kS KHS Roughness height of Hull surface m Frequency, commonly rate of rev- n N Hz olution PB PB Brake power Power delivered by prime mover W Brake power in representative sea 𝑃 PBW W BW condition PD , PP PD, PP Delivered power, propeller power Q ω W Delivered Power, corrected using 𝑃 PDT 𝑃 𝐶 ⋅𝑃 W correlation factor PE , PR PE, PR Effective power, resistance power R V W Determined from pressure meas- P PI Indicated power W I ured by indicator PS PS Shaft power Power measured on the shaft W PT PTH Thrust power T VA W Q Q Torque PD / ω Nm Full scale resistance without over- 𝑅 R0 N load

𝑅Tw RTW Total resistance in wind and waves N tV TV Running trim m V V Ship speed m/s Equivalent propeller open water VA VA Propeller advance speed speed based on thrust or torque m/s identity Design ship speed when the ship is 𝑉ref VREF in operation in a calm sea condi- m/s tion (no wind and waves) Design ship speed when the ship is 𝑉w VW in operation under the representa- m/s tive sea condition zV ZV Running sinkage of model or ship m ω V0,OMN Rotational shaft velocity 2 π n 1/s 2.3.2.2 Derived Quantities a RAUG Resistance augment fraction (T - RT) / RT 1

𝐵f BF Bluntness coefficient 7.5-04-01-01.1 1 2/3 3 CADM CADM Admiralty coefficient Δ V / PS 1 3 2/3 CD CDVOL Power-displacement coefficient PD / (ρ V  / 2) 1 Trial correction for propeller rate C CN n / n 1 N of revolution at speed identity T S Trial correction for propeller rate C CNP P / P 1 NP of revolution at power identity DT DS Trial correction for delivered C CDP 1 P power ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.2 Ship Performance 55

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Weather factor, a non-dimensional speed in wind and waves 𝑓 coefficient indicating the decrease speed in calm water 𝑓 FWE 1 𝑤 of speed in representative sea con- 𝑉 ditions 𝑉 Ship model correlation factor for K C1 η / η 1 1 propulsive efficiency DS DM Ship model correlation factor for K C2 n / n 1 2 propeller rate revolution S M Scale effect correction factor for model appendage drag applied at K KAP Appendage correction factor 1 APP the towing force in a self-propul- sion test Change of draught, fore and aft, s SINKV Sinkage, dynamic 1 V divided by length Change of the trim due to dynamic t TRIMV Trim, dynamic 1 V condition, divided by length t THDF Thrust deduction fraction (T - RT) / T 1 w WFT Taylor wake fraction in general (V - VA) / V 1 wF WFF Froude wake fraction (V - VA) / VA 1 Propeller speed V determined w WFTQ Torque wake fraction A 1 Q from torque identity Effect of the rudder(s) on the wake 𝑤 1 R fraction Propeller speed, V , determined w WFTT Thrust wake fraction A 1 T from thrust identity

𝛥𝑅𝑤𝑎𝑣𝑒𝑠 DRWA Added wave resistance N

𝛥𝑅𝑤𝑖𝑛𝑑 DRWI Added wind resistance N Ship-model correlation factor for Δw DELW w - w 1 wake fraction T,M T,S Ship-model correlation factor with ΔwC DELWC respect to wT,S method formula of 1 ITTC 1978 method x XLO Load fraction in power prediction ηD PD / PE - 1 1 Ship appendage resistance divided β APSF Appendage scale effect factor 1 by model appendage resistance Load variation coefficient of the 𝜉 1 shaft revolution speed Load variation coefficient of the 𝜉 1 delivered power Load variation coefficient of the 𝜉 1 ship speed 2.3.2.3 Efficiencies etc. ηAPP ETAAP Appendage efficiency PEw0APP / PEwAPP, RTBH / RT 1 ETAB, η Propeller efficiency behind ship P / P = T V / (Q ω) 1 B EFTP T D A ETAD, Quasi-propulsive efficiency coef- η P / P = P / P 1 D EFRP ficient E D R P Propulsive efficiency in ideal con- 𝜂 ETADID 1 Did dition, from model test

ETAG, η Gearing efficiency 1 G EFGP ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.2 Ship Performance 56

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

ETAH, P / P = P / P η Hull efficiency E T R T 1 H EFRT = (1 - t) / (1 - w) Mechanical efficiency of transmis- η ETAM P / P 1 M sion between engine and propeller D B ETAO, P / P = T V / (Q ω) all quanti- η Propeller efficiency in open water T D A 1 O EFTPO ties measured in open water tests ηP ETAP Propulsive efficiency coefficient PE / PB 1 ETAR, η Relative rotative efficiency η / η 1 R EFRO B O ETAS, η Shafting efficiency P / P = P / P 1 S EFPS D S P S ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.3 Propulsor Performance 57

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.3.3 Propulsor Performance 2.3.3.1 Basic Quantities 2 2 AO AO Propeller disc area π D / 4 m D DP Propeller diameter m n FR Propeller frequency of revolution Hz Propeller rate of revolution, cor- 𝑛 𝑛 𝐶 ⋅𝑛 1 rected using correlation factor Roughness height of propeller k KS m P blade surface Dynamic pressure based on ad- q QA ρ V 2 / 2 Pa A vance speed A Dynamic pressure based on sec- q QS ρ V 2 / 2 Pa S tion advance speed S About spindle axis of controllable QS QSP Spindle torque pitch propeller QS=QSC+ QSH Nm positive if it increases pitch QSC QSPC Centrifugal spindle torque Nm QSH QSPH Hydrodynamic spindle torque Nm RU RU Pod unit resistance Resistance of a podded drive unit N T TH Propeller thrust N Pod unit resistance subtracted T TU Pod unit thrust N U from the propeller thrust Duct thrust of a ducted propeller T THDU N D unit Propeller thrust of a ducted propel- T THDP N P ler unit Total thrust of a ducted propeller T THDT N T unit TxP TXP Propeller Thrust along shaft axis N Propeller normal force in y direc- T TYP N yP tion in propeller axis Propeller normal force in z direc- T TZP N zP tion in propeller axis VA VA Advance speed of propeller m/s Mean axial velocity at propeller V VP m/s P plane of ducted propeller Section advance speed V VS (V 2+ (0.7 R ω)2)1/2 m/s S at 0.7 R A 𝑌 YU Pod unit side force N The portion of the resistance (load Resistance fraction for one propel-  fraction,  ) that the ith propeller is 1 i ler i responsible for 3 ρP DNP Propeller mass density kg/m Thrust deduction sensitivity for 𝛥𝐹 i 𝜏 1 1 one propeller 𝛥𝑇 i ω V0P Propeller rotational velocity 2 π n 1/s 2.3.3.2 Derived Quantities ½ 2.5 n PD / VA Taylor's propeller coefficient with n in revs/min, BP BP 1 based on delivered horsepower PD in horsepower, and VA in kn (obsolete) ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.3 Propulsor Performance 58

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

½ 2.5 n PT / VA Taylor's propeller coefficient with n in revs/min, BU BU 1 based on thrust horsepower PT in horsepower, and VA in kn (obsolete) CP CPD Power loading coefficient PD / (AP qA VA) 1 CQ* CQS Torque index Q / (AP qS D) 1 Thrust loading coefficient, energy C CTH T / (A q ) = (T / A ) / q 1 Th loading coefficient P A P P A CT* CTHS Thrust index T / (AP qS) 1 J JEI Propeller advance ratio VA / (D n) 1 JA , JH JA, JH Apparent or hull advance ratio V / (D n) = VH / (D n) 1 Propeller advance ratio for ducted J JP V / (D n) P propeller P Advance ratio of propeller deter- J , J JT, JPT 1 T PT mined from thrust identity Advance ratio of propeller deter- J , J JQ, JPQ 1 Q PQ mined from torque identity 3 5 KP KP Delivered power coefficient PD / (ρ n D ) = 2 π KQ 1 2 5 KQ KQ Torque coefficient Q / (ρ n D ) 1 Centrifugal spindle torque coeffi- K KSC Q / (ρ n2 D5) 1 SC cient SC Hydrodynamic spindle torque co- K KSH Q / (ρ n2 D5) 1 SH efficient SH 2 4 KT KT Thrust coefficient T / (ρ n D ) 1 Duct thrust coefficient for a ducted K KTD T / (ρ n2 D4) 1 TD propeller unit D Propeller thrust coefficient for a K KTP T / (ρ n2 D4) 1 TP ducted propeller unit P Thrust coefficient achieved by 𝐾 KTQ 1 torque identity Total thrust coefficient for a K KTT K + K 1 TT ducted propeller unit TP TD Torque coefficient of propeller . KQ0 KQ0 converted from behind to open KQ ηR 1 water condition Torque coefficient of propeller de- KQT KQT termined from thrust coefficient 1 identity PJ PJ Propeller jet power ηTJ T VA SA SRA Apparent slip ratio 1 - V / (n P) 1 SR SRR Real slip ratio 1 - VA / (n P) 1 n D / VA δ ADCT Taylor's advance coefficient with n in revs/min, D in feet, VA 1 in kn Propeller pump or hydraulic effi- η EFJP P / P = P / P 1 JP ciency J D J P Propeller pump efficiency at zero ZET0, η advance speed, T / (ρ π / 2)1/3 / (P D)2/3 1 JP0 EFJP0 D alias static thrust coefficient ηI EFID Ideal propeller efficiency Efficiency in non-viscous fluid 1 1/2 ηTJ EFTJ Propeller jet efficiency 2 / (1 + (1 + CTh) ) 1 ETA0, P / P = T V / (Q ω) all quanti- η , η Propeller efficiency in open water T D A 1 0 TP0 EFTP0 ties measured in open water tests λ ADR Advance ratio of a propeller VA / (n D) / π = J / π 1 ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.3 Propulsor Performance 59

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Ratio between propeller thrust and τ TMR T / T 1 total thrust of ducted propeller P T 2.3.3.3 Induced Velocities etc. Axial velocity induced by propel- U UA m/s A ler Axial velocity induced by duct of U UADU m/s AD ducted propeller Radial velocity induced by propel- U URP m/s RP ler of ducted propeller Radial velocity induced by duct of U URDU m/s RD ducted propeller Axial velocity induced by propel- U UAP m/s AP ler of ducted propeller Radial velocity induced by propel- U UR m/s R ler Tangential velocity induced by U UTDU m/s TD duct of ducted propeller Tangential velocity induced by U UTP m/s TP propeller of ducted propeller Tangential velocity induced by U UT m/s T propeller Advance angle of a propeller blade β BETB arctg (V / r ω) rad section A Hydrodynamic flow angle of a Flow angle taking into account in- β BETI rad I propeller blade section duced velocity * β BETS Effective advance angle arctg (VA/ (0.7 R ω)) rad

ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.4 Unsteady Propeller Forces 60

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.3.4 Unsteady Propeller Forces 2.3.4.1 Basic Quantities Cuv SI(U,V) Generalized stiffness Duv DA(U,V) Generalized damping u = 1,.., 6 N Generalized vibratory F FG(I) u = 1, 2, 3: force N u force u = 4, 5, 6: moment Nm Fi F(I) Vibratory force i = 1, 2, 3 N Generalized vibratory force coeffi- According to definitions of KFi KFu KF(U) 1 cients and KMi 2 4 KFi KF(I) Vibratory force coefficients Fi / (ρ n D ) 1 2 5 KMi KM(I) Vibratory moment coefficients Mi / (ρ n D ) 1 2 2 Kp KPR Pressure coefficient p / (ρ n D ) 1 Mi M(I) Vibratory moment i = 1, 2, 3 Nm Muv MA(U,V) Generalized mass p PR Pressure Pa u = 1,.., 6 N Generalized vibratory bearing re- R R(U) u = 1, 2, 3: force N u action u = 4, 5, 6: moment Nm Vi V(I) Velocity field of the wake i = 1, 2, 3 m/s Origin O coinciding with the cen- tre of the propeller. The longitudi- x X m nal x-axis coincides with the shaft y Y Cartesian coordinates m axis, positive forward; the trans- z Z m verse y-axis, positive to port; the third, z-axis, positive upward Cylindrical system with origin O and longitudinal x-axis as defined X X before; angular a-(attitude)-coordi- m a ATT Cylindrical coordinates nate , zero at 12 o'clock position, 1 r R positive clockwise looking for- m ward; r distance measured from the x-axis u = 1,.., 6 m Generalized vibratory displace-  DP(U) u = 1, 2, 3: linear m u ment u = 4, 5, 6: angular rad u = 1,.., 6 m/s DPVL(U) Generalized vibratory velocity u = 1, 2, 3: linear m/s  u u = 4, 5, 6: angular rad/s u = 1,.., 6 m/s2 DPAC(U) Generalized vibratory acceleration u = 1, 2, 3: linear m/s2 u u = 4, 5, 6: angular rad/s2

ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.5 Water Jets 61

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.3.5 Water Jets Cp CP Local pressure coefficient (p-p0)/(ρ V²/2) 1

Thrust loading coefficient:viscous Tnet C 1 Tn pressure 1 2 2 UA0n Energy velocity coefficient at sta- c 1 es tion s Momentum velocity coefficient at c 1 ms station s Pressure differential of flow rate Dp Pa transducer

2 Ej EJ Energy flux at station j Ej = (ρ/2)VEj dQj W QJ

Total energy flux at station s (ki-  1 2 p  Es  2 u gxjj undA ii W netic + potential + pressure)    As  

Total axial (in  direction) energy  1 2 p  Es  2 ugxundA jj ii W flux at station s    As   Skin friction correction in a self F propulsion test carried out at the N D ship self-propulsion point H1 HT1 Local total head at station 1 m Mean increase of total head across H35 H35 pump and stator or several pump m stages IVR IVR Intake velocity ratio VI/V 1 JVR JVR Jet velocity ratio VJ/V 1 Q K Impeller torque coefficient: Q nD25 Q K Flow rate coefficient: J 1 QJ nD3 Momentum flux at station s in i di- uundAijj  N M is rection  As u NVR Nozzle velocity ratio: 6 1 U0 Jet thrust (can be measured di- T TJX N jx rectly in bollard pull condition) n Impeller rotation rate Hz n Unit normal vector in i direction 1 i P Delivered Power to pump impeller W D P Effective power: RU W E TBH 0 P Effective Jet System Power QH W JSE J1A7 P Pump effective power: QH W PE J35

PTE Effective thrust power W ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.5 Water Jets 62

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Ambient pressure in undisturbed p PR0 Pa 0 flow p Local static pressure at station s Pa s Q Impeller torque Nm Volume flow rate inside boundary Q m³/s bl layer Volume flow rate through water Q m³/s J jet system R Total resistance of bare hull N TBH Jet thrust (can be measured di- T N jet x rectly in bollard pull condition) Net thrust exerted by the jet sys- T N net tem on the hull

RTBH t Thrust deduction fraction 1t 1 Tnet U Free stream velocity m/s 0

Mean energy velocity in i direction 1 3 udA m/s uesi at station s   QJ

Mean (total) energy velocity at sta- 1 3 u es u dA m/s tion s Q  J Velocity component in i-direction u m/s is at station s

us Velocity at station s m/s Local tangential velocity at station u UJFI m/s 7φ 7 w Geometric intake width at station 1 m 1 Width of capture area measured w m 1A over hull surface at station 1A Vertical distance of nozzle centre z m 6 relative to undisturbed surface ΔM DMF Change of momentum flux N Change in Momentum Flux in x N M x direction

PE  Overall propulsive efficiency: 1 D PD

PJSE  Ducting efficiency: 1 duct PPE

PJSE0  Energy interaction efficiency: 1 eI PJSE Ideal efficiency, equivalent to jet PTE0  efficiency in free stream condi- 1 I tions PJSE0 Installation efficiency to account  for the distorted flow delivered by 1 inst the jet intake to the pump ITTC Symbols 2 Ships in General 2.3 Resistance and Propulsion Version 2021 2.3.5 Water Jets 63

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

eI  Total interaction efficiency: 1 t 1 INT  mI

PTE  Momentum or jet efficiency: 1 jet PJSE

PJSE  Jet system efficiency: 1 JS PD

Tnet0  Momentum interaction efficiency: 1 mI Tnet

PPE  ETAP Pump efficiency 1 P PD Pump efficiency from a pump loop  1 P0 test  Free stream efficiency:   1 0 P duct I Jet angle relative to the horizontal  rad n at the nozzle (station 6)  Mass density of fluid kg/m³ Energy loss coefficient between  1 ij station i and j

E31 E  ZETA13 Inlet duct loss coefficient: 1 13 1 2 2 U0

EE75  ZETA57 Nozzle duct loss coefficient: 2 1 57 1 2 ue6

ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.1 Manoeuvrability 64

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.4 Manoeuvrability and Seakeeping 2.4.1 Manoeuvrability 2.4.1.1 Geometrical Quantities see also Section 1.3.1 and Section 1.3.3 2 AFB AFBO Projected area of bow fins m The area of the profile of the un- derwater hull of a ship when pro- A AHLT Lateral area of the hull m2 HL jected normally upon the longitu- dinal centre plane 2 ALV AHLV Lateral area of hull above water m 2 AR ARU Total lateral area of rudder m Lateral area of movable part of A ARMV m2 Rmov rudder 2 ARN ARNO Nominal lateral area of rudder (AR + ARmov) / 2 m Maximum distance from root to b SPRU Rudder span m R tip bRM SPRUME Mean span of rudder m CAL CAHL Coefficient of lateral area of ship AHL / (L T) 1 h DE Water depth m hM DEME Mean water depth m Longitudinal position of rudder x XRU m R axis δ ANRU Rudder angle, helm angle rad 2 ΛR ASRU Aspect ratio of rudder bR / AR 1 2.4.1.2 Motions and Attitudes Roll velocity, rotational velocity p OX, P 1/s about body x-axis Pitch velocity, rotational velocity q OY, Q 1/s about body y-axis Yaw velocity, rotational velocity r OZ, R 1/s about body z-axis Roll acceleration, angular acceler- p OXRT, PR dp / dt 1/s2  ation about body x-axis Pitch acceleration, angular accel- q OYRT, QR dq / dt 1/s2  eration about body y-axis Yaw acceleration, angular acceler- OZRT, RR dr / dt 1/s2 r ation about body z-axis Surge velocity, linear velocity u UX, U m/s along body x-axis Sway velocity, linear velocity v UY, V m/s along body y-axis Heave velocity, linear velocity w UZ, W m/s along body z-axis Surge acceleration, linear acceler- UXRT, UR du / dt m/s2 u ation along body x-axis Sway acceleration, linear accelera- UYRT, VR dv / dt m/s2 v tion along body y-axis Heave acceleration, linear acceler- UZRT, WR dw / dt m/s2 w ation along body z-axis Linear velocity of origin in body V V m/s axes VA,V0 VA, V0 Approach speed m/s Vu V(URT) Generalized velocity m/s ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.1 Manoeuvrability 65

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

V(URT) Generalized acceleration m/s2 V u VF VF Flow or current velocity m/s VWR VWREL Relative wind velocity m/s VWT VWABS True wind velocity m/s ψ YA Course angle or heading rad χ YX Yaw angle rad dtψ YART Rate of change of course dψ / dt rad/s ΨO YA0R Original course rad θ PI Pitch angle rad  RO Roll angle rad 2.4.1.3 Flow Angles etc. α AAPI Pitch angle Angle of attack in pitch on the hull rad β AADR Drift angle Angle of attack in yaw on the hull rad βWR ANWIRL Angle of attack of relative wind rad Angle of a control surface, rudder δ ANCS rad angle, helm angle δ0 ANRU0 Neutral rudder angle rad δEFF ANRUEF Effective rudder inflow angle rad δFB ANFB Bow fin angle rad δFS ANFS Stern fin angle rad δR ANRU Rudder angle rad δRO ANRUOR Rudder angle, ordered rad ψC COCU Course of current velocity rad ψWA COWIAB Absolute wind direction see also section 3.4.2, Wind rad ψWR COWIRL Relative wind direction rad

2.4.1.4 Forces and Derivatives Roll moment on body, moment K MX Nm about body x-axis Pitch moment on body, moment M MY Nm about body y-axis Yaw moment on body, moment N MZ Nm about body z-axis Derivative of yaw moment with Nr NR Nms respect to yaw velocity Nr/ 

Derivative of yaw moment with 2 N r NRRT Nms  respect to yaw acceleration N /r Derivative of yaw moment with Nv NV Ns respect to sway velocity Nv/ 

Derivative of yaw moment with 2 N v NVRT Nms  respect to sway acceleration Nv/   Derivative of yaw moment with Nδ ND Nm respect to rudder angle N /  QFB QFB Torque of bow fin Nm QR QRU Torque about rudder stock Nm QFS QFS Torque of stern fin Nm Surge force on body, force along X FX N body x-axis XR XRU Longitudinal rudder force N Derivative of surge force with re- Xu XU Ns/m spect to surge velocity Xu/  ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.1 Manoeuvrability 66

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

X u Derivative of surge force with re-  XURT Xu/  Ns2/m spect to surge acceleration  Sway force, force in direction of Y FY N body axis y Derivative of sway force with re- Yr YR Ns spect to yaw velocity Yr/  YR YRU Transverse rudder force N Y r Derivative of sway force with re-  YRRT Yr/  Ns2 spect to yaw acceleration  Derivative of sway force with re- Yv YV Ns/m spect to sway velocity Yv/ 

Derivative of sway force with re- 2 Y v YVRT Ns /m  spect to sway acceleration Yv/  Derivative of sway force with re- Yδ YD N spect to rudder angle Y /  Heave force on body, force along Z FZ N body z-axis 2.4.1.5 Linear Models 2 2 Cr CRDS Directional stability criterion Yv (Nr - muxG) - Nv (Yr - mu) N s Lb , lb LSB Static stability lever Nv / Yv m Ld , ld LSR Damping stability lever (Nr - muxG) / (Yr - mu) m Time constant of the 1st order T TIC s manoeuvring equation First time constant of manoeu- T TIC1 s 1 vring equation Second time constant of manoeu- T TIC2 s 2 vring equation Third time constant of manoeu- T TIC3 s 3 vring equation Gain factor in linear manoeuvring K KS 1/s equation P-number, heading change per unit P PN 1 n rudder angle in one ship length 2.4.1.6 Turning Circles DC DC Steady turning diameter m Non-dimensional steady turning D ’ DCNO D / L 1 C diameter C PP Inherent steady turning diameter D0 DC0 m δR = δ0 Non-dimensional inherent steady D  DC0N D0 / LPP 1 0 turning diameter Loop height of r-δ curve for unsta- l LHRD rad/s r ble ship Loop width of r-δ curve for unsta- l LWRD rad δ ble ship rC OZCI Steady turning rate 1/s Non-dimensional steady turning r  OZCINO rC LPP / UC or 2 LPP / DC m C rate RC RCS Steady turning radius m Time to reach 90 degree change of t TI90 s 90 heading Time to reach 180 degree change t TI180 s 180 of heading ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.1 Manoeuvrability 67

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

UC UC Speed in steady turn m/s x090 X090 Advance at 90 change of heading m Advance at 180 change of head- x X0180 m 0180 ing x0max XMX Maximum advance m y 090 Y090 Transfer at 90 change of heading m

Tactical diameter (transfer at 180 y Y0180 m 0180 change of heading) y0max Y0MX Maximum transfer m βC DRCI Drift angle at steady turning rad 2.4.1.7 Zig-Zag Manoeuvres ta TIA Initial turning time s tc1 TIC1 First time to check yaw (starboard) s tc2 TIC2 Second time to check yaw (port) s thc TCHC Period of changes in heading s tr TIR Reach time s y0max Y0MX Maximum transverse deviation m δmax ANRUMX Maximum value of rudder angle rad ψS PSIS Switching value of course angle rad ψ01 PSI01 First overshoot angle rad ψ02 PSI02 Second overshoot angle rad 2.4.1.8 Stopping Manoeuvres Distance along track, s SPF m F track reach x0F X0F Head reach m y0F Y0F Lateral deviation m tF TIF Stopping time s

ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.2 Seakeeping 68

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.4.2 Seakeeping Related information is to be found in Chapter 3 on General Mechanics in Sections 3.1.2 on Time and Frequency Domain Quantities, 3.1.3 on Stochastic Processes, 3.2.1 on Inertial Properties,, 3.2.2 on Loads, 3.2.3 on Rigid Body Motions, and 3.4.1 on Waves. 2.4.2.1 Basic Quantities 2 AFS AFS Projected area of stern fins m Attitudes of the floating i = 1, 2, 3, e.g. Euler angles of roll, a AT(I) rad i system pitch, and yaw, respectively f FR Frequency 1 / T Hz fE FE Frequency of wave encounter 1 / TE Hz fz Natural frequency of heave 1 / Tz Hz fθ Natural frequency of pitch 1 / Tθ Hz fφ Natural frequency of roll 1 / Tφ Hz FL FS(2) Wave excited lateral shear force Alias horizontal! N FN FS(3) Wave excited normal shear force Alias vertical! N MB(3), Wave excited lateral bending mo- M Alias horizontal! Nm L FS(6) ment MB(2), Wave excited normal bending mo- M Alias vertical! Nm N FS(5) ment MT(1), M Wave excited torsional moment Nm T FS(4) Mean increased rate of revolution n NAW 1/s AW in waves PAW PAW Mean power increased in waves W QAW QAW Mean torque increased in waves Nm Mean resistance increased in R RAW N AW waves S (f), S (f), EWSF, Wave elevation auto spectral den- η ηη see also section 1.4.1, Waves m2s Sη(ω), Sηη(ω) EWSC sity Absolute displacement of the ship i = 1, 2, 3 :surge, sway, x X(I) m i at the reference point and heave respectively Generalized displacement of a u = 1...6 surge, sway, heave, roll, x X(U) m, rad u ship at the reference point pitch, yaw TAW TAW Mean thrust increase in waves N T TC Wave period s Te TE Wave encounter period s Tz TNHE Natural period of heave s Tθ TNPI Natural period of pitch s Tφ TNRO Natural period of roll s Y (ω), Amplitude of frequency response z (ω) / ζ (ω) or z a a 1 Azζ(ω) function for translatory motions za(ω) / ηa(ω) Yθζ(ω), Amplitude of frequency response Θa(ω) / ζa(ω) or 2 1 Aθζ(ω) function for rotary motions Θa(ω) / (ω / (gζa(ω)))

wwwE EE LLLz ===qf

wwwZ qj Λ Tuning factor or T TTZ q j LLLZ ===qf TTTE EE Angle between ship positive x axis and positive direction of waves μ Wave encounter angle rad (long crested) or dominant wave direction (short crested) ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.2 Seakeeping 69

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.3 Large Amplitude Motions Capsizing 70

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

2.4.3 Large Amplitude Motions Capsizing Assumed centre of gravity above A keel used for cross curves of sta- 1 bility - I99/1.2.4.1 Longitudinal centre of buoyancy Distance of centre of buoyancy XAB from aft perpendicular - m AB from aft perpendicular I99/1.2.4.2 AC Area of deck available to crew m² Distance of the centre of flotation XAF m AF from after perpendicular Longitudinal centre of gravity Distance of centre of gravity from XAG m AG L from aft perpendicular aft perpendicular Transverse distance from assumed AG T YAG centre of gravity A, to actual cen- m tre of gravity G Vertical distance from assumed AGV ZAG centre of gravity A, to actual cen- m tre of gravity G ALV AHLV Lateral area of hull above water m² Positive area under righting lever A m² RL curve A SI ASI Attained subdivision index 1 IAS Area of sails in profile according A AS m² S to ISO 8666 Projected lateral area of the por- tion of the ship and deck cargo A AV m² V above the waterline - IMO/IS, IMO/HSC'2000 Righting arm based on horizontal Generally tabulated in cross YAZ distance from assumed centre of m AZ curves of stability gravity A, to Z Centroid of the underwater vol- B Centre of buoyancy ume Breadth between centres of buoy- B m CB ancy of side hulls Distance from the centre of buoy- ancy CB to transverse metacentre Transverse metacentre above cen- M BM ZBM m tre of buoyancy I BM  T  KM  KB  Longitudinal metacentre above ZBML m BM L centre of buoyancy BM L  KM L  KB Centre of flotation of added buoy- b ancy layer or centre of lost buoy- ancy of the flooded volume b Maximum tank breadth m Proportion of boat plan needed for C Crew density D crew Height coefficient, depending on the height above sea level of the C 1 H structural member exposed to the wind ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.3 Large Amplitude Motions Capsizing 71

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Maximum number of persons on C Crew limit Lcpi board Trimming moment divided by Longitudinal trimming coefficient change in trim which approxi- C CMTL 1 MTL - I99/1.2.4.3 mately equals BML / L Shape coefficient, depending on Cs the shape of the structural member 1 exposed to the wind Draught, moulded, of ship hull - d T m I99/1.2.1 Density coefficient for submerged d 1 test weights Centre of flotation of the water F plane F Wind force - IMO/IS From the freeboard markings to f FREB Freeboard the freeboard deck, according to m official rules Longitudinal centre of buoyancy, Distance of centre of buoyancy XFB m FB LCB, from forward perpendicular from forward perpendicular Longitudinal centre of flotation, Distance of centre of floatation XFF m FF LCF, from forward perpendicular from forward perpendicular Longitudinal centre of gravity, Distance of centre of gravity from XFG m FG from forward perpendicular forward perpendicular G Centre of gravity of a vessel Centre of gravity of an added or g 1 removed weight (mass) Vertical stability lever caused by a GGV m GG1 weight shift or weight addition KG 1  KG 0  GG 1 Horizontal stability lever caused GGH GGH by a weight shift or weight addi- m tion Longitudinal stability lever caused GG L GGL by a weight shift or weight addi- m tion Vertical stability lever caused by a GGV m GGV weight shift or weight addition KG 1  KG 0  GG1 Distance of centre of gravity to the metacentre GM GM Transverse metacentric height GM  KM  KG m (not corrected for free surface ef- fect) Effective transverse metacentric GM Corrected for free surface GM EFF GMEFF m height and/or free communication effects Distance from the centre of gravity G to the longitudinal metacentre GML GML Longitudinal metacentric height ML m

GM L  KM L  KG

GZ  AZ  AGV sin - GZ GZ Righting arm or lever m AG T cos φ ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.3 Large Amplitude Motions Capsizing 72

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Arm of static stability corrected m GZ for free surfaces - IMO/table

GZ MAX GZMAX Maximum righting arm or lever m h Maximum tank height m Height of centre of area of A h SP m CE above waterline at SSM Heeling lever (due to various rea- HL m sons) - IMO/HSC'2000 Height of waterline above centre h m LP of area of immersed profile K Keel reference Distance from the assumed centre Assumed centre of gravity above ZKA of gravity A to the moulded base m KA moulded base of keel of keel or K Distance from the centre of buoy- Centre of buoyancy above ZKB ancy B to the moulded base of keel m KB moulded base of keel or K Centre of gravity above moulded Distance from the centre of gravity ZKG m KG base of keel G to the moulded base of keel or K Vertical centre of gravity of added Distance from the assumed centre Kg ZKAG or removed weight above moulded of gravity, g, to the moulded base m base of keel of keel or K Distance from the transverse meta- Transverse metacentre above ZKM centre M to the moulded base of m KM moulded base of keel keel or K Distance from the longitudinal Longitudinal metacentre above ZKML metacentre M to the moulded m KM L moulded base of keel L base of keel or K Roll damping coefficient express- k 1 ing the effect of bilge keels Length of the vessel on the water- L line in maximum load condition - m IMO/IS Arm of dynamic stability cor- l rected for free surfaces - m IMO/table

l XTA Longitudinal trimming arm XXCG CB m l Maximum tank length m Actual length of enclosed super- ls structure extending from side to m side of the vessel lw Wind heeling lever m M Metacentre of a vessel See subscripts for qualification m SHIPMA Ship mass W/g kg Maximum offset load moment due M Nm C to crew Minimum capsizing moment as Mc determined when account is taken Nm of rolling Free surface moment at any incli- M Nm FS nation mLCC Mass in light craft condition kg ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.3 Large Amplitude Motions Capsizing 73

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Mass in loaded displacement con- m kg LDC dition according to … mMTL Maximum total load (mass) kg MR Heeling moment due to turning Nm  GZ . Other moments such as Moment of ship stability in gen- M MS those of capsizing, heeling, etc. Nm S eral will be represented by MS with ad- ditional subscripts as appropriate. Mass in standard sailing condi- m kg SSC tions according to … Moment to change trim one centi- M MTC Nm/cm TC metre

MTM MTM Moment to change trim one meter  CMTL Nm/m Maximum heeling moment due to M Nm W wind Dynamically applied heeling mo- M Nm v ment due to wind pressure Height of centre of gravity above m OG waterline PV Wind pressure Pa r Effective wave slope coefficient 1 RSI RSI Required subdivision index 1 s Wave steepness 1 Actual stability index value ac- STIX STIX 1 cording to … Required stability index value, see STIXR 1 STIX … T YHA Equivalent transverse heeling arm Heeling moment/ m TL Turning lever m ts TRIM Static trim TA - TF - dKL m tKL V Tank total capacity m³ v Speed of craft in the turn - V0 IMO/HSC'2000 m/s Service speed - IMO/IS vW Wind speed used in calculation m/s W SHIPWT Ship weight m g N Longitudinal distance from refer- Longitudinal centre of floatation x XACB ence point to the centre of the m CB of added buoyant layer added buoyant layer, b Longitudinal distance from refer- X Longitudinal centre of buoyancy CB XCB ence point to the centre of buoy- m L (L ) CB CB ancy, B Longitudinal distance from refer- X Longitudinal centre of flotation CF XCF ence point to the centre of flota- m L (L CF CF) tion, F Longitudinal distance from refer- Longitudinal centre of gravity of ence point to the centre of gravity, x XACG m CG added weight (mass) g, of an added or removed weight (mass) ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.3 Large Amplitude Motions Capsizing 74

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Longitudinal distance from refer- X Longitudinal centre of gravity CG XCG ence point to the centre of gravity, m L (L ) CG CG G X1, X2 Roll damping coefficients 1 Distance of down flooding open- x m D ing from end of boat Y Lateral displacement of centre of Lateral distance from a reference CG, YCG m yCG gravity (YCG) point to the centre of gravity, G Distance of down flooding open- y m D ing from gunwale Distance of down flooding open- y ' m D ing off centreline Intersection of righting arm with Z ZRA line of action of the centre of buoyancy Vertical distance from the centre of A to the centre of the underwa- Z ter lateral area or approximately to m a point at one half the draught - IMO/IS Vertical distance from the centre Z, h m of A to the waterline Height above waterline of down z m D flooding opening zSA ZSA Static sinkage at AP Caused by loading m zSF ZSF Static sinkage at FP Caused by loading m zS ZS Mean static sinkage (zSF+zSA)/2 m  Tank block coefficient 1 tKL DTR Change in static trim m  DISPF Displacement (buoyant) force g  N m DISPM Displacement mass  kg  DISPVOL Displacement volume g m³ Displacement volume of flooded  DISVOLFW  g m³ fw water fw  HEELANG Heel angle rad  Heel angle during offset load tests rad Maximum permitted heel angle   rad (REQ) during … Actual down flooding angle ac-  rad D cording to … Required down flooding angle,   rad D(REQ) see… Down flooding angle to non-quick  rad DC draining cockpits Down flooding angle to any main  rad DH access hatchway

F HEELANGF Heel angle at flooding rad Angle of heel at which maximum   rad GZMAX righting moment occurs

R Assumed roll angle in a seaway rad VS HEELANGV Heel angle for vanishing stability rad W Heel angle due to calculation wind rad ITTC Symbols 2 Ships in General 2.4 Manoeuvrability and Seakeeping Version 2021 2.4.3 Large Amplitude Motions Capsizing 75

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

The ratio of the volume of flood-  PMVO Volumetric permeability ing water in a compartment to the 1 total volume of the compartment Capsizing angle under the action   rad c of a gust of wind IMO/IS Heel angle corresponding to the m maximum of the statical stability rad curve -1 S TRIMS Static trim angle tan ((zSF-zSA)/L) rad  RHO (Liquid) mass density kg/m³ RHOA  (Air) mass density kg/m³ DNA

 DNWA (Water) mass density kg/m³ 2.4.4 Symbols for Attributes and Subscripts A Aft E Entrance F Fore R Run Z Heave θ Pitch φ Roll ITTC Symbols 3 Special Craft 3.1 Planing and Semi-Displacement Vessels Version 2021 3.1.1 Geometry and Hydrostatics 76

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3. SPECIAL CRAFT 3.1 Planing and Semi-Displacement Vessels 3.1.1 Geometry and Hydrostatics See also Section 1.2.1, Hull Geometry and Section 1.2.2 Propulsor Geometry Horizontally projected planing 2 AP APB Planing bottom area bottom area (at rest), excluding m area of external spray strips Breadth over spray strips meas- Breadth at longitudinal position of B BLCG ured at transverse section contain- m LCG the centre of gravity ing centre of gravity Breadth over chines, excluding ex- B BPC Breadth over chines m PC ternal spray strips BPA BPA Mean breadth over chines AP / LP m Breadth over chines at transom, B BPT Transom breadth m PT excluding external spray strips Maximum breadth over chines, ex- B BPX Maximum breadth over chines m PX cluding external spray strips LSB LSB Total length of shafts and bossings m Length of chine projected in a L LPRC Projected chine length m PR plane parallel to keel Angle between a straight line ap- proximating body section and the β BETD Deadrise angle of planing bottom rad intersection between basis plane and section plane βM BETM Deadrise angle at midship section rad βT BETT Deadrise angle at transom rad Angle between shaft line and ref- εSH EPSSH Shaft angle erence line (positive, shaft inclined rad downwards) ITTC Symbols 3 Special Craft 3.1 Planing and Semi-Displacement Vessels Version 2021 3.1.2 Geometry and Levers, Underway 77

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.1.2 Geometry and Levers, Underway 3.1.2.1 Geometry, Underway Vertical depth of trailing edge of dTR DTRA Immersion of transom, underway boat at keel below water surface m level Wetted height of strut palms h HSP m P (flange mounting) hR HRU Wetted height of rudders m LC LC Wetted chine length, underway m Lever of resultant of pressure Distance between centre of pres- l LCP m CP forces, underway sure and aft end of planing surface LK LK Wetted keel length, underway m LM LM Mean wetted length, underway (LK + LC) / 2 m Principal wetted area bounded by Wetted area underway of planing S SWHP trailing edge, chines and spray m2 WHP hull root line Area bounded by stagnation line, 2 SWB SWB Wetted bottom area, underway chines or water surface underway m and transom Total wetted surface of hull un- derway, including spray area and S SWHE Wetted hull area, underway m2 WHE wetted side area, w/o wetted tran- som area Wetted area of the hull side above S SWSH Area of wetted sides m2 WHS the chine or the design water line Wetted area between design line S , S SWS Area wetted by spray m2 WS S or stagnation line and spray edge Angle between projected keel and stagnation line in a in plane nor- α ALFSL Angle of stagnation line rad B mal to centre plane and parallel to reference line Angle between barrel axis and as- α ALFBAR Barrel flow angle rad BAR sumed flow lines εWL EPSWL Wetted length factor LM / LWL 1 εWS EPSWS Wetted surface area factor S / S0 1 Angle between design waterline Running trim angle based on de- θ , TRIMDWL and running waterline (positive rad DWL sign waterline bow up) Angle between ship design water- line and actual water line at rest θS, θ0 TRIMS Static trim angle -1 rad (positive bow up) tan ((zSF - zSA) / L) Angle between actual water line at rest and running water line (posi- θ , θ TRIMV Running (dynamic) trim angle rad V D tive bow up) -1 tan ((zVF - zVA) / L) λW LAMS Mean wetted length-breadth ratio LM / (BLCG) 1 Angle between design waterline Running trim angle based on de-  TRIMDWL and running waterline (positive deg sign waterline bow up) Angle between the reference line τ TAUDWL Reference line angle rad DWL and the design waterline τ Angle of attack relative to the ref- Angle between the reference line R TAUR rad erence line and the running waterline ITTC Symbols 3 Special Craft 3.1 Planing and Semi-Displacement Vessels Version 2021 3.1.2 Geometry and Levers, Underway 78

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Angle between stagnation line and φ PHISP Spray angle rad SP keel (measured in plane of bottom) Effective increase in friction area Dimensionless increase in total δ DLAM length-breadth ratio due to spray 1 λ friction area contribution to drag 3.1.2.2 Levers, Underway Distance between NA and centre of eA ENAPP Lever of appendage lift force NA m gravity (measured normally to NA) Distance between NB and centre of eB ENBOT Lever of bottom normal force NB m gravity (measured normally to NB) Distance between propeller centre- Lever of propeller normal force e ENPN line and centre of gravity (meas- m PN N PN ured along shaft line) Distance between N and centre Lever of resultant of propeller PP ePP ENPP of gravity (measured normally to m pressure forces NPP NPP) Distance between N and centre Lever of resultant propeller suc- PS ePS ENPS of gravity (measured normal to m tion forces NPS NPS) Distance between N and centre Lever of resultant of rudder pres- RP eRP ENRP of gravity (measured normal to m sure forces NRP NRP) Distance between RAA and centre fAA FRAA Lever of wind resistance RAA of gravity (measured normal to m RAA) Distance between RAP and centre fAP FRAP Lever of appendage drag RAP of gravity (measured normal to m RAP) Distance between RF and centre of fF FRF Lever of frictional resistance RF m gravity (measured normal to RF) Distance between RK and centre of fK FRK Lever of skeg or keel resistance RK m gravity (measured normal to RK) Distance between ΔR and centre Lever of augmented rudder drag RP fR FDRR of gravity (measured normal to m ΔRRP ΔRRP) Distance between axial thrust and fS FSL Lever of axial propeller thrust centre of gravity (measured nor- m mal to shaft line) Distance between RT and centre of fT FRT Lever of total resistance RT m gravity (measured normal to RT) ITTC Symbols 3 Special Craft 3.1 Planing and Semi-Displacement Vessels Version 2021 3.1.3 Resistance and Propulsion 79

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.1.3 Resistance and Propulsion See also Sections 2.3.1 on Hull Resistance 2 CL0 CL0D Lift coefficient for zero deadrise Δ / (BCG q) 1 Lift coefficient for deadrise sur- C CLBET Δ / (B 2 q) 1 Lβ face CG 1/2 CV CSP Froude number based on breadth V / (BCG g) 1 3 CΔ CDL Load coefficient Δ / (BCG ρ g ) 1 Vertical component of hydrody- L LVD N VHD namic lift LVS LVS Hydrostatic lift Due to buoyancy N Drag forces arising from append- Appendage drag force (parallel to F FTAPP ages inclined to flow, assumed to N TA reference line) act parallel to the reference line Viscous component of bottom Bottom frictional force (parallel to F FTBOT drag forces assumed acting paral- N TB reference line) lel to the reference line Drag forces arising from keel or Keel or skeg drag force (parallel to F FTKL skeg, assumed to act parallel to the N TK reference line) reference line Drag forces arising from influence Additional rudder drag force (par- of propeller wake on the rudder F FTRP N TRP allel to reference line) assumed to act parallel to the ref- erence line Lift forces arising from append- Appendage lift force (normal to N NAPP ages inclined to flow, assumed to N A reference line) act normally to reference line Resultant of pressure and buoyant Bottom normal force (normal to N NBOT forces assumed acting normally to N B reference line) the reference line Resultant of propeller pressure Propeller pressure force (normal to N NPP forces acting normally to the refer- N PP reference line) ence line Resultant of propeller suction Propeller suction force (normal to N NPS forces acting normally to the refer- N PS reference line) ence line Resultant of rudder pressure forces Rudder pressure force (normal to N NRP acting normally to the reference N RP reference line) line RK RKEEL Keel drag N Rπ RPI Induced drag g ρ  tan τ N Drag due to inlet and outlet open- R RPAR Parasitic drag N PAR ings RPS RSP Pressure component of spray drag N RT RT Total resistance Total towed resistance N RVS RSV Viscous component of spray drag CF SWS qS N Mean velocity over bottom of the V VBM Mean bottom velocity m/s BM hull Relative velocity between hull and V VSP Spray velocity m/s SP spray in direction of the spray ITTC Symbols 3 Special Craft 3.2 Multi-Hull Vessels Version 2021 3.2.1 Geometry and Hydrostatics 80

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.2 Multi-Hull Vessels 3.2.1 Geometry and Hydrostatics See also Section 2.2.1, Hull Geometry 2 AI AIA Strut-hull intersection area m BB BB Box breadth Breadth of main deck m BS BS Hull spacing Distance between hull centre lines m Minimal distance of the demihulls B BTUN Tunnel width m TV at the waterline Diameter of axis symmetric sub- D DHUL Hull diameter m H merged hulls Hull diameter at the longitudinal D DX m X position "X" Minimum clearance of wet deck H HCLDK Deck clearance m DK from water surface at rest Depth of strut from still water line H HSS Strut submerged depth m SS to strut-hull intersection Half angle of entrance at tunnel Angle of inner water line with ref- i ANENIN rad EI (inner) side erence to centre line of demihull Half angle of entrance at outer Angle of outer water line with ref- i ANENOU rad EO side erence to centre line of demihull LCH LCH Length of centre section of hull Length of prismatic part of hull m LCS LCS Length of centre section of strut Length of prismatic part of strut m LH LH Box length Length of main deck m Length of nose section of hull with L LNH Length of nose section of hull m NH variable diameter Length of nose section of strut L LNS Length of nose section of strut m NS with variable thickness Length of strut from leading to L LS Strut length m S trailing edge LSH LSH Length of submerged hull m tS TSTR Maximum thickness of strut m ITTC Symbols 3 Special Craft 3.2 Multi-Hull Vessels Version 2021 3.2.2 Resistance and Propulsion 81

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.2.2 Resistance and Propulsion 3.2.2.1 Resistance Components See also Section 2.3.1 on Hull Resistance Frictional resistance of multi-hull R RFMH N FMH vessel Frictional resistance interference R RFINT R - Σ R N FINT correction FMH F Residuary resistance correction of R RRMH R - R N RMH multi-hull TMH FMH Residuary resistance interference R RRINT R - Σ R N RI correction RMH R Total resistance of multi-hull ves- R RTMH N TMH sel Total resistance interference cor- R RTINT R - Σ R N TI rection TMH T

ITTC Symbols 3 Special Craft 3.3 Hydrofoil Boats Version 2021 3.3.1 Geometry and Hydrostatics 82

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.3 Hydrofoil Boats 3.3.1 Geometry and Hydrostatics See Sections 2.2.1 and 2.2.4 2 AF AFO Foil area (general) Foil area in horizontal plane m 2 AFT AFT Total foil plane area m Maximum vessel breadth includ- B BFOA m FOA ing foils bS BST Span of struts m Transverse horizontal distance of b BSTT m ST struts cC CHC Chord length at centre plane m cF CFL Chord length of flap m cM CHM Mean chord length m cS CSTR Chord length of a strut m Chord length of strut at intersec- c CHSF m SF tion with foil cT CHTI Chord length at foil tips m WF WTF Weight of foil N αc ALFTW Geometric angle of twist rad θDH DIHED Dihedral angle rad 3 F DISVF Foil displacement volume m 3.3.1.1 Geometry, Underway 2 AFE AFE Emerged area of foil m 2 AFF ASFF Submerged area of front foil m 2 AFR ASFR Submerged area of rear foil m 2 AFS AFS Submerged foil area m Submerged foil plan area at take- A AFSTO m2 FST0 off speed 2 ASS ASS Submerged strut area m bw BSPW Foil span wetted m Distance of centre of pressure on a c CPFL m PF foil or flap from leading edge Froude number based on foil dis- Fr FNFD V / (g L )1/2 1 L tance F Froude number based on chord Fr FNC V / (g c )1/2 1 c length M Height of centre of gravity foil- Distance of centre of gravity h HVCG m CG borne above mean water surface Height of foil chord at foilborne h HFL Flight height m F mode above position at rest Distance between keel and mean h HKE Keel clearance m K water surface foilborne Horizontal distance of centre of lF LEFF pressure of front foil to centre of m gravity Horizontal distance between cen- lFR LEFR tres of pressure of front and rear lF + lR m foils Horizontal distance of centre of lR LERF pressure of rear foil to centre of m gravity Distance between foil chord and T TFO Foil immersion m F mean water surface ITTC Symbols 3 Special Craft 3.3 Hydrofoil Boats Version 2021 3.3.1 Geometry and Hydrostatics 83

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

Depth of submergence of apex of Distance between foil apex and T TFD m FD a dihedral foil mean water surface TFM TFOM Mean depth of foil submergence m αIND ALFIND Downwash or induced angle rad Angle of attack of mean lift coeffi- α ALFM rad M cient for foils with twist Angle of attack for which flow α AFS rad s separation (stall) occurs αTO ATO Incidence angle at take-off speed rad ITTC Symbols 3 Special Craft 3.3 Hydrofoil Boats Version 2021 3.3.2 Resistance and Propulsion 84

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.3.2 Resistance and Propulsion See also Section 2.3.1 Hull Resistance 3.3.2.1 Basic Quantities Force in the direction of motion of D DRF Foil drag N F an immersed foil DFR DFA Drag force on rear foil CDF AFR q N DFF DFF Drag force on front foil CDF AFF q N For finite span foil, the component D DRIND Induced drag N I of lift in the direction of motion Due to mutual interaction of the DINT DRINT Interference drag boundary layers of intersecting N foil Profile drag for angle of attack D DRF0 Streamline drag N P0 equal to zero lift DS DRSP Spray drag Due to spray generation N DST DRST Strut drag N Due to propagation of surface D DRWA Wave drag N W waves Due to reduced pressure at the rear D DRVNT Ventilation drag N V side of the strut base LF LF Lift force on foil CL AFT q N LFF LFF Lift force on front foil CL AFF q N LFR LFR Lift force on rear foil CL AFR q N Profile lift force for angle of attack L LF0 C A q N 0 of zero L0 FT LTO LT0 Lift force at take off CLTO AFT q N M MSP Vessel pitching moment Nm ITTC Symbols 3 Special Craft 3.3 Hydrofoil Boats Version 2021 3.3.2 Resistance and Propulsion 85

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.3.2.2 Derived Quantities CDF CDF Drag coefficient of foil DF / (AFS q) 1 CDI CDI Induced drag coefficient DI / (AFS q) 1 CDINT CDINT Interference drag coefficient DINT / (AFS q) 1 Section drag coefficient for angle C CDO D / (A q) 1 D0 of attack equal to zero P FS CDS CDSP Spray drag coefficient DS / (AFS q) 1 CDVENT CDVENT Ventilation drag coefficient DV / (AFS q) 1 CDW CDW Wave drag coefficient DW / (AFS q) 1 CLF CLF Foil lift coefficient LF / (AFS q) 1 Profile lift coefficient for angle of C CLO L / (A q) 1 L0 attack equal to zero 0 FS Lift coefficient at take-off condi- C CLTO L / (A q) 1 LTO tion TO FS CLX CLA Slope of lift curve dCL / dα 1 CM CM Pitching moment coefficient M / ((AFF + AFR) (lF - lR) q) 1 MF MLF Load factor of front foil LFF / Δ 1 MR MLR Load factor of rear foil LFR / Δ 1 εF EPSLDF Lift/ Drag ratio of foil L / D 1 ITTC Symbols 3 Special Craft 3.4 ACV and SES Version 2021 3.4.1 Geometry and Hydrostatics 86

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.4 ACV and SES 3.4.1 Geometry and Hydrostatics See also Section 1.2.1 Projected area of ACV or SES A CUA Cushion area m2 C cushion on water surface SES cushion breadth measured be- B BCU Cushion breadth m C tween the side walls BWLT BWLT Total waterline breadth of SES At the water line m Height of centre of gravity above H HVCG m CG mean water plane beneath craft Distance from side wall keel to h HBS Bow seal height m BS lower edge of bow seal HSK HSK Skirt depth m Distance from side wall keel to h HSS Stern seal height m SS lower edge of stern seal LB LB Deformed bag contact length m LC LAC Cushion length m LE LACE Effective length of cushion AC / BC m Wetted area of side hulls at rest off Total wetted area of side walls un- S SSH0 m2 H0 cushion der way on cushion Wetted area of side hulls under Total wetted area of side walls un- S SSHC m2 SHC way on cushion der way on cushion Wetted area of side hulls under Total wetted area of side walls un- S SSH m2 SH way off cushion der way off cushion Horizontal spacing between inner XH, LH XH, LH and outer side skirt hinges or at- needs clarification m tachment points to structure Distance of leading skirt contact XS, LS XS, LS point out-board or outer hinge of needs clarification m attachment point to structure Vertical spacing between inner ZH, HH ZH, HH and outer side skirt hinges or at- needs clarification m tachment points to structure Increase in cushion breadth due to δB DBCV m C water contact εWS EPSWS Wetted surface factor SSHC / SSH0 1 θB TETB Bag contact deformation angle rad θF TETF Finger outer face angle rad Slope of mean water plane for sur- θW TETW face level beneath cushion periph- rad ery (ACV and SES) Mass density of ρ DNA Mass of air per unit volume kg/m3 A air Height of cushion generated wave ζC ZETAC above mean water plane at leading m edge side of the skirt ITTC Symbols 3 Special Craft 3.4 ACV and SES Version 2021 3.4.1 Geometry and Hydrostatics 87

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.4.2 Resistance and Propulsion See also Section 2.3.1 on Hull Resistance 3/2 CΔ CLOAD Cushion loading coefficient Δ / (g ρA AC ) 1 Aerodynamic profile drag coeffi- C CPR R / (ρ V 2 A / 2) 1 PR cient 0 A R C CWC CWC Cushion wave making coefficient 1 pB PBM Mean bag pressure Pa pBS PBS Bow seal pressure Pressure in the bow seal bag Pa pCE PCE Mean effective skirt pressure Pa pCU PCU Cushion pressure Mean pressure in the cushion Pa pFT PFT Fan total pressure Pa pLR PLR Cushion pressure to length ratio PCU / LC Pa/m pSK PSK Skirt pressure in general Pa pSS PSS Stern seal pressure Pressure in the stern seal bag Pa PFCU PFCU Power of lift fan W PFSK PFSK Power of skirt fan W 3 QBS QBS Bow seal air flow rate Air flow rate to the bow seal m /s 3 QCU QCU Cushion air flow rate Air flow rate to cushion m /s 3 QSS QSS Stern seal air flow rate Air flow rate to the stern seal m /s 3 QT QT Total air volume flow m /s 3 QTS QTS Total air volume flow of skirt m /s RAT RAT Total aerodynamic resistance RM + R0 N RH RH Hydrodynamic resistance RW + RWET N Intake momentum resistance in R RM ρ Q V N M general A T A Intake momentum resistance of R RMCU ρ Q V N MCU cushion A CU A Intake momentum resistance of R RASK ρ Q V N ASK skirt A TS A RWET RWET Resistance due to wetting N TC TC0 Cushion thrust N ITTC Symbols 3 Special Craft 3.5 Ice going Vessels Version 2021 3.5.1 Resistance and Propulsion 88

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.5 Ice Going Vessels

3.5.1 Resistance and Propulsion (See Figure 3.4, p 225 and Figure 3.8, p 231 of Vol. 1 of the Proceedings of the 21st ITTC) 2 CI CI Coefficient of net ice resistance RI / (ρI g h B) 1 Coefficient of water resistance in C CIW R / (S q ) 1 IW the presence of ice IW IW Projection of hull - ice interaction F FNIC Normal ice force on a body N IN force on the external normal Projection of the hull - ice interac- FIT FTIC Tangential ice force on a body tion force on the direction of mo- N tion Froude number based on ice thick- Fr FNIC V / (g h )1/2 1 I ness I FXI FXIC N FYI FYIC Components of the local ice force N FZI FZIC N Ratio of tangential force to normal Coefficient of friction between f CFRD force between two bodies (dy- 1 ID surface of body and ice (dynamic) namic condition) Coefficient of friction between The same as above (static condi- f CFRS 1 IS surface of body and ice (static) tion) hI HTIC Thickness of ice m hSN HTSN Thickness of snow cover m Average coefficient of torque in K KQICMS Q / (ρ n 2 D5) 1 QIA ice IA W IA 2 4 KTIA KTICMS Average coefficient of thrust in ice TIA / (ρW nIA D ) 1 Average rate of propeller revolu- n FRICMS Hz IA tion in ice PDI PDI Delivered power at propeller in ice 2 π QIA nIA W QIA QIMS Average torque in ice Nm RI RI Net ice resistance RIT - RIW N RIT RIT Total resistance in ice Ship towing resistance in ice N Hydrodynamic resistance in pres- Total water resistance of ship in R RIW N IW ence of ice ice TIA TIMS Average total thrust in ice N Relative propulsive efficiency in η ERIC η / η 1 ICE ice ID D ηID EFDIC Propulsive efficiency in ice RIT V / (2 π nIA QIA) 1 ITTC Symbols 3 Special Craft 3.6 Sailing Vessels Version 2021 3.6.1 Geometry and Hydrostatics 89

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.6 Sailing Vessels 3.6.1 Geometry and Hydrostatics See also Section 2.2.1 on Hull Geometry 2 AJ ASJ Area of jib or genoa m 2 ALK ALK Lateral area of keel m 2 ALT ALT Total lateral area of yacht m 2 Am ASM Area of mainsail m 2 AN ASN Normalized sail area m 2 ASP ASSP Area of spinnaker m 2 AS, SA AS Sail area in general (P E + I J) / 2 m BOA BOA Breadth, overall m C CPI Center of pressure for A Main- pi i m E EM sail base I I Fore triangle height m J J Fore triangle base m P P Mainsail height m Effective length for Reynolds L LEFF m EFF Number Wetted surface area of canoe S SC m2 C body 2 SK SK Wetted surface area of keel m SR SR Wetted surface area of rudder m² TC TCAN Draught of canoe body m 2 5 TEFF TEFF Effective draught FH / (ρVB R) m Height of centre of effort of sails ZCE ZCE above waterline in vertical cen- m tre plane 3 C DVCAN Displaced volume of canoe body m 3 K DVK Displaced volume of keel m 3 R DVR Displaced volume of rudder m Displacement force (weight) of Δ DFCAN N C canoe body Displacement force (weight) of Δ DFK N K keel Displacement force (weight) of Δ DFR N R rudder ITTC Symbols 3 Special Craft 3.6 Sailing Vessels Version 2021 3.6.2 Resistance and Propulsion 90

ITTC Computer Definition or SI- Name Symbol Symbol Explanation Unit

3.6.2 Resistance and Propulsion Frictional resistance coefficient C CFU R / (S q) 1 FU (upright) FU Residuary resistance coefficient C CRU R / (S q) 1 RU (upright) RU Total resistance coefficient (up- C CTU R / (S q) 1 TU right) TU Wave resistance coefficient (up- C CWU 1 WU right) Total resistance coefficient with C CTPHI R / (S q) 1 Tφ heel and leeway Tφ CI Induced resistance coefficient 1 Cx, Cy, Cz Force coefficients 1 FH Heeling force of sails N FR Driving force of sails N FV Vertical force of sails N H Side force N LHY Hydrodynamic lift force N RAW Mean added resistance in waves N RFU Friction resistance (upright) N RRU Residuary resistance (upright) N Resistance increase due to side R N I (induced resistance) RTU RTU Total resistance (upright) N RTφ RTUH Total resistance when heeled RTU + Rφ N Resistance increase due to heel R , R RTUHA N φ H (with zero side force) Components of resultant force X,Y,Z N along designated axis V V Vessel velocity m/s VWR VWR Apparent wind velocity m/s VWT VWT True wind velocity m/s Vmc VMC Velocity made good on course m/s Velocity made good to windward V VMG m/s mg (contrary to wind direction) βL BETAL leeway angle rad apparent wind angle (relative to β BETWA rad aw boat course) true wind angle (relative to boat β BETWT rad tw course)

ITTC Symbols 4 Background and references

Version 2021 91

4. BACKGROUND AND REFERENCES far from being developed and understood to the 4.1 Symbols and Terminology Group extent necessary. A consequence of this situa- The tasks of the former Symbols and Termi- tion is that the symbols proposed are not always nology Group (SaT) have been handed over to as coherent as would be necessary for advanced the Quality Systems Group in 2002. and systematic work, for which explicit models and adequate notations are essential. 4.2 Description of the List of Symbols The problem under discussion is of course 4.2.1 Classification the same in national and international standards. The prime concern of the QS Group was to However there is an accepted international revise and try to complement the list of ITTC standard which deals with the general principles Standard Symbols sticking to the system for the concerning physical quantities, equations, quan- classification of concepts. tity and unit symbols, and coherent unit systems With this regard, the following design re- for general use within the various fields of sci- quirements and goals have been maintained: ence and technology (ISO 31. 4.2.3 Organization of the matrix structured 1. a coherent document, meeting the present and possibly the future requirements of the ITTC list community in general and particular user As has been emphasized the development of groups symbols is a continuing process and as the sub- ject develops, further amendments and additions, 2. an open ended matrix structure that can be eas- as approved by the Conference, will be included ily expanded as requirements arise, without the need of restructuring and repetition or too many in future editions of the list. explicit cross-references In order to avoid any extra problems the symbols are arranged in alphabetical order in 3. minimized departures from the well established each subject area as in previous lists. Continu- and widely accepted previous list of symbols ous page numbering was discarded in earlier On the other hand, to facilitate the practical versions. The idea was to establish a loose leaf use of the list, a second version in which the organization as the most appropriate, in view of symbols are arranged in alphabetic order was new draughts to be incorporated. prepared. Symbols which have been listed sev- In view of the tremendous effort which ex- eral times in the matrix structured document plicit mathematical models, explanations, and have been maintained and for each symbol the sketches take for their preparation, the present field in which it is used is given in italic letters QS Group can only follow the former SaT prior to the meaning of the symbol. Group and state that the Technical Committees 4.2.2 Structure of the Lists and other interested parties are urged to provide The concepts related to a given subject area further material for review by the QS Group and or model are designated by the ITTC Symbol future inclusion into the list. and called by their Name. Their meaning can in It has been noted that some users dislike the principle only be concluded from the context of disruption of the list of symbols by lengthy ex- the model. The logically consistent, so called planations. The present QS Group feels that the 'implicit' definition is derived from a definitely subject and the sensible use of the symbols re- defined statement of the model, ideally a gener- quire such explanations, also as the fundamen- ally accepted system or an equivalent, e.g. a tals of the theory of science and terminology of- drawing. ten are not taught to students of naval architec- The problem is that traditionally in lists of ture and marine engineering. However the ar- symbols, as in dictionaries, these explicit mod- rangement has been changed so that these expla- els are missing for various reasons. One reason nations can be visited by using hyperlinks and is that many subject areas under discussion are the list is not disrupted any more.

ITTC Symbols 5 Principles of notation

Version 2021 92

5. PRINCIPLES OF NOTATION  exponentiation In Figure 5 the principles of notation in ac-  the various aspects of complex quantities cording to ISO 31 are shown.  the various aspects of spectra and Symbols representing physical quantities  the various aspects of random quantities and sto- chastic processes e.g. probability operators. normally are one Latin or Greek letter with Sub- scripts for further identification. They are writ- Subscripts signify identifiers ten in italic style letters.  matrix components, Numbers are normally written in roman  identifiers tested, e.g. ship S or model M, append- style letters. For more details, look at the list be- ages (App)or the various bodies in a multi-body problem, low or in the Excerpts of ISO 31 below or in  identifiers of coordinate systems and of the refer- standard itself. ence points, quantities( LPP) Superscripts signify operators e.g.

Symbols for physical units italic, one letter, except dimensionless quanti- A ( e.g. Area in m²) ties Symbols for characteristic numbers 2 letters italic Re, Fr Numbers roman, generally 103 Symbols representing numbers italic xij Units roman, lower case unless derived from name m, Pa Prefix of units roman µm Symbols for chemical elements roman H2O Symbols for universal constants italic g = 9,80665 m/s²

ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 93

Superscripts signify Operators Numbers in roman (power of n) Symbols representing numbers (n = 1, 2, 3 etc.) in italic

SYMBOLS 2 n NUMBERS in italic KTM 10 in roman Subscripts signify Identifiers: Symbol for physical quantity in italic (T = Thrust) Other symbols in roman (M = Model) Figure 5

5.1 Excerpts of ISO 31 so on would constitute such a category. Mutually compa- rable quantities are called "quantities of the same kind". 1 Scope If a particular example of a quantity from such a cat- This part of ISO 31 gives general infor- egory is chosen as a reference quantity, called the unit, then any other quantity from this category can be ex- mation about principles concerning physical pressed in terms of this unit as a product of this unit and a quantities, equations, quantity and unit symbols, number. This number is called the numerical value of and coherent unit systems, especially the Inter- the quantity expressed in this unit. national System of Units, SI. In formal treatments of quantities and units, The principles laid down in this part of ISO this relation may be expressed in the form 31 are intended for general use within the vari- A={A}-[A] ous fields of science and technology and as a where A is the symbol for the physical quantity, general introduction to the other parts of ISO 31. [A] the symbol for the unit and {A} symbolizes the numerical value of the quantity A expressed 2. Quantities and units in the unit [A]. For vectors and tensors the com- 2.1 Physical quantity, unit and numerical value In ISO 31 only physical quantities used for the quan- ponents are quantities which may be expressed titative description of physical phenomena are treated. as described above. Conventional scales, such as the Beaufort scale, Richter If a quantity is expressed in another unit scale and colour intensity scales, and quantities expressed which is k times the first unit, then the new nu- as the results of conventional tests, e.g. corrosion re- merical value becomes 1/k times the first numer- sistance, are not treated here, neither are currencies nor information contents. ical value; the physical quantity, which is the Physical quantities may be grouped together into cat- product of the numerical value and the unit, is egories of quantities which are mutually comparable. thus independent of the unit. Lengths, diameters, distances, heights, wavelengths and ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 94

REMARK ON NOTATION FOR NUMERICAL VALUES 2.2.4 Numerical factors in quantity equations It is essential to distinguish between the Equations between quantities sometimes contain nu- quantity itself and the numerical value of the merical factors. These numerical factors depend on the definitions chosen for the quantities occurring in the equa- quantity expressed in a particular unit. The nu- tions. merical value of a quantity expressed in a par- EXAMPLE ticular unit could be indicated by placing braces 1 Ek = mv² (curly brackets) around the quantity symbol and 2 using the unit as a subscript. It is, however, pref- erable to indicate the numerical value explicitly 2.2.5 Systems of quantities and equations between as the ratio of the quantity to the unit. quantities; base quantities and derived quantities Physical quantities are related to one another through 2.2 Quantities and equations equations that express laws of nature or define new quanti- 2.2.1 Mathematical operations with quantities ties. Two or more physical quantities cannot be added or For the purpose of defining unit systems and introduc- subtracted unless they belong to the same category of mu- ing the concept of dimensions, it is convenient to consider tually comparable quantities. some quantities as mutually independent, i.e. to regard Physical quantities are multiplied or divided these as base quantities, in terms of which the other quan- by one another according to the rules of algebra; tities can be defined or expressed by means of equations; the latter quantities are called derived quantities. the product or the quotient of two quantities, A It is a matter of choice how many and which quantities and B, satisfies the relations are considered to be base quantities. AB = {A} {B} - [A] [B] The whole set of physical quantities included in ISO 31 Thus, the product {A} {B} is the numerical value is considered as being founded on seven base quantities: {AB} of the quantity AB, and the product [A] [B] is the length L, mass M, time T, electric current I, thermody- unit [AB] of the quantity AB. Similarly, the quotient namic temperature Θ, amount of substance N and lu- {A!/{B}) is the numerical value {A/B} of the quantity minous intensity J. A/B, and the quotient [A]/[B] is the unit [A/B] of the In the field of mechanics a system of quantities and quantity A/B. equations founded on three base quantities is generally used. In ISO 31-3, the base quantities used are length, mass 2.2.2 Equations between quantities and equations be- and time. tween numerical values In the field of electricity and magnetism a system of Two types of equation are used in science and technol- quantities and equations founded on four base quantities is ogy: equations between quantities, in which a letter generally used. In ISO 31-5, the base quantities used are symbol denotes the physical quantity (i.e. numerical value length, mass, time and electric current. × unit), and equations between numerical values. In the same field, however, systems founded on only Equations between numerical values depend on the choice three base quantities, length, mass and time, in particular of units, whereas equations between quantities have the ad- the "Gaussian" or symmetric system, have been widely vantage of being independent of this choice. Therefore the used. (See ISO 31-5:1992, annex A.) use of equations between quantities should normally be preferred. 2.2.6 Dimension of a quantity Any quantity Q can be expressed in terms of other 2.2.3 Empirical constants quantities by means of an equation. The expression may An empirical relation is often expressed in the form of consist of a sum of terms. Each of these terms can be ex- an equation between the numerical values of certain physi- pressed as a product of powers of base quantities A, B, C, cal quantities. Such a relation depends on the units in which ... from a chosen set, sometimes multiplied by a numerical the various physical quantities are expressed. factor ξ, i.e. ξAαBβCγ ..., where the set of exponents (α, β, γ An empirical relation between numerical values can be ...) is the same for each term. transformed into an equation between physical quantities, The dimension of the quantity Q is then expressed by containing one or more empirical constants. Such an equa- the dimensional product tion between physical quantities has the advantage that the dim Q = AαBβCγ .. form of the equation is independent of the choice of the units. The numerical values of the empirical constants oc- where A, B, C, ... denote the dimensions of the base quan- curring in such an equation depend, however, on the units tities A, B, C, ..., and where α, β, γ .... are called the di- in which they are expressed, as is the case with other phys- mensional exponents. ical quantities. ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 95

A quantity all of whose dimensional exponents are General Conference on Weights and equal to zero is often called a dimensionless quantity. Its Measures (Conférence Générale des Poids et dimensional product or dimension is A0 B0 C0... = 1. Such a quantity of dimension one is expressed as a number. Mesures, CGPM) in 1960. This system includes In the system founded on the seven base quantities - base units length, mass, time, electric current, thermodynamic tem- - derived units including supplementary units perature, amount of substance and luminous intensity, the which together form the coherent system of SI units. base dimensions may be denoted by L, M, T, I, O, N and J respectively and the dimension of a quantity Q becomes in general 2.3.2.1 Base units dim Q = LαMβTγIδΘεNζJη. The seven base units are listed in Table 1.

EXAMPLES Table 1 - SI base units SI base unit Base quantity Quantity Dimension Name Symbol velocity LT-1 angular velocity T-1 length metre m force LMT-2 mass kilogram kg 2 -2 energy L MT time second s relative density 1 electric current ampere A thermodynamic tempera- kelvin K 2.3 Units ture 2.3.1 Coherent unit systems amount of substance mole mol Units might be chosen arbitrarily, but making an inde- luminous intensity candela cd pendent choice of a unit for each quantity would lead to the appearance of additional numerical factors in the equations between the numerical values. 2.3.2.2 Derived units including supplementary units It is possible, however, and in practice more conven- ient, to choose a system of units in such a way that the equa- The expressions for the coherent derived units in terms tions between numerical values have exactly the same form of the base units can be obtained from the dimensional (including the numerical factors) as the corresponding products by using the following formal substitutions: equations between the quantities. A unit system defined in L → m I → A this way is called coherent with respect to the system of M → kg Θ → K quantities and equations in question. The SI is such a sys- T → s N →mol tem. The corresponding system of quantities is given in ISO J → cd 31-1 to ISO 31-10 and in ISO 31-12 and ISO 31-13. In 1960, the CGPM classified the SI units radian, rad, For a particular system of quantities and equations, a and steradian, sr, for plane angle and solid angle respec- coherent system of units is obtained by first defining units tively as "supplementary units". for the base quantities, the base units. Then for each de- In 1980, the International Committee for Weights and rived quantity, the definition of the corresponding derived Measures (Comité International des Poids et Mesures, unit in terms of the base units is given by an algebraic ex- CIPM) decided to interpret the class of supplementary units pression obtained from the dimensional product (see 2.2.6) in the SI as a class of dimensionless derived units for which by replacing the symbols for the base dimensions by those the CGPM allows the freedom of using or not using them of the base units. In particular, a quantity of dimension one in expressions for SI derived units. acquires the unit 1. In such a coherent unit system no nu- Although, as a consequence of this interpretation, the merical factor other than the number 1 ever occurs in the coherent unit for plane angle and for solid angle is the num- expressions for the derived units in terms of the base units. ber 1, it is convenient to use the special names radian, rad, and steradian, sr, instead of the number 1 in many practical 2.3.2 SI units and their decimal multiples and cases. sub-multiples The name International System of Units (Système International d'Unités), with the inter- national abbreviation SI was adopted by the 11th

Table 2 - SI derived units with special names, including SI supplementary units ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 96

SI derived unit Derived quantity Expressed in terms of SI base Special name Symbol units and SI derived units plane angle radian rad 1 rad = 1 m/m = 1 solid angle steradian sr 1 sr = 1 m2 /m2 = 1 frequency hertz Hz 1 Hz = 1 s-' force newton N 1 N = 1 kg . m/s2 pressure, pascal Pa 1 Pa = 1 N/m2 stress energy, joule J 1 J = 1 N - m work, quantity of heat power, watt W 1 W = 1 J/s radiant flux electric charge, coulomb C 1 C = 1 A - s quantity of electricity electric potential, volt V 1 V = 1 W/A potential difference, tension, electromotive force capacitance farad F 1 F = 1 C/V electric resistance ohm S2 1Ω = 1 V/A electric conductance siemens S 1 S = 1 Ω-' magnetic flux weber Wb 1 Wb = 1 V . s magnetic flux density tesla T 1 T = 1 Wb/m2 inductance henry H 1 H = 1 Wb/A Celsius temperature degree °C 1 °C = 1 K Celsius') luminous flux lumen Im 1 lm = 1 cd . sr illuminance lux Ix 1 lx = 1 lm/m2 1) Degree Celsius is a special name for the unit kelvin for use in stating values of Celsius temperature. (See also ISO 31-4:1992, items 4-1.a and 4-2.a.)

For some of the SI derived units, special names and EXAMPLES symbols exist; those approved by the CGPM are listed in Quantity Symbol for SI unit tables 2 (and 3). expressed in terms of It is often of advantage to use special names and symbols the seven base units (and in compound expressions for units. the supplementary units in some cases) 2.3.2.3 SI prefixes

velocity m/s In order to avoid large or small numerical values, dec- angular velocity rad/s or s-' imal multiples and sub-multiples of the SI units are added force kg . m/s2 to the coherent system within the framework of the SI. energy kg . m2/s2 They are formed by means of the prefixes listed in Table relative density 1 4.

ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 97

NOTES Table 4 - Sl prefixes 3 In some countries the symbol %o ("per mill", or per thou- Prefix Factor sand) is used for the number 0,001. This symbol should be Name Symbol avoided. 1024 yotta Y 4 Since per cent and per mill are numbers it is 1021 zetta z in principle meaningless to speak about percentage by mass 1018 exa E or percentage by volume. Additional information, such as 1015 peta P % (m/m) or % (V/V), should not therefore be attached to 1012 tera T the unit symbol. The preferred way of expressing a mass 109 giga G fraction is: "the mass fraction is 0,67" or "the mass fraction 106 mega M is 67 %", and the preferred way of expressing a volume 103 kilo k fraction is: "the volume fraction is 0,75" or "the volume 102 hecto h fraction is 75 %". Mass and volume fractions can also be 10 deca da expressed in the form 5 µg/g or 4,2 ml/m3. 10-1 deci d Abbreviations such as ppm, pphm and ppb shall not be 10-2 centi c used. 10-3 milli m 10-6 micro µ 2.3.4 Other unit systems and miscellaneous 10-9 nano n units 10-12 pico p The CGS system of mechanical units is a coherent sys- 10-15 femto f tem the base units of which are centimetre, gram and sec- 10-18 atto a ond for the three base quantities length, mass and time. 10-21 zepto z In practice this system was enlarged by adding the kel- 10-24 yocto y vin, the candela and the mole as base units for the base quantities thermodynamic temperature, luminous intensity For information about the use of the prefixes, see and amount of substance. 3.2.4. Units used in electricity and magnetism have been de- fined in the CGS system in several ways depending on the The SI units and their decimal multiples and submulti- system of quantities and equations chosen. The "Gaussian" ples formed by use of the prefixes are specially recom- or symmetric CGS system, coherent with the "Gaussian" or mended. symmetric system of quantities and equations founded on three base quantities, has been widely used. For further in- 2.3.3 The unit one formation on this system, see ISO 31-5:1992, Annex A. The coherent SI unit for any quantity of dimension one is the unit one, symbol 1. It is generally not written The special names and symbols for derived CGS units out explicitly when such a quantity is expressed numeri- such as dyne, erg, poise, stokes, gauss, oersted and maxwell cally. shall not be used together with the Sl. EXAMPLE Table 5 - Units used with the SI

Refractive index n = 1,53 × 1 = 1,53 Quantity Unit Name Symbol Definition In the case of certain such quantities, however, the time minute min 1 min= 60s unit 1 has special names that could be used or not, depend- hour h 1 h = 60 min ing on the context. day d 1 d = 24 h

EXAMPLES plane angle degree ° 1 ° = (π/180)rad = Plane angle α = 0,5 rad minute ' (n/l80) rad = 0,5 , second '' 1' = (1/60)° Solid angle Ω = 2,3 sr 1" = (1/60)' = 2,3

Decimal multiples and sub-multiples of the unit one volume litre I, L 1) 1l = 1 dm3 are expressed by powers of 10. They shall not be ex- = 1 dm pressed by combining the symbol 1 with a prefix. mass tonne2) t 1 t = 103 kg In some cases the symbol % (per cent) is used for the 1) The two symbols for litre are on an equal footing. The number 0,01. CIPM will, however, make a survey on the development ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 98

of the use of the two symbols in order to see if one of the 5 Symbols for quantities are given in ISO 31-1 to ISO 31- two may be suppressed. 10 and in ISO 31-12 and ISO 31-13. 2) Also called the metric ton in the English language. 6 Notations for vectorial and other non-scalar quantities Table 6 - Units used with the SI, whose values in SI units are given in ISO 31-11, on mathematical signs and sym- are obtained experimentally bols.

Unit 7 Exceptionally, symbols made up of two letters are Quantity sometimes used for combinations of dimension one of Name Symbol Definition quantities (e.g. Reynolds number, Re). If such a two-letter energy electronvolt eV symbol appears as a factor in a product, it is recommended that it be separated from the other symbols. The electronvolt is the kinetic energy acquired 3.1.2 Rules for the printing of subscripts by an electron in passing When, in a given context, different quantities have the through a potential dif- same letter symbol or when, for one quantity, different ap- ference of 1 volt in vac- plications or different values are of interest, a distinction uum: 1 eV  can be made by use of subscripts. 1,602 177 ×10-19 J. The following principles for the printing of subscripts are recommended: A subscript that represents a symbol for a physical mass unified u The unified atomic mass quantity is printed in italic (sloping) type. atomic unit is equal to (1 /12) of Other subscripts are printed in roman (upright) type. mass unit the mass of an atom of the nuclide 12C: 1u  EXAMPLES 1,660 540 ×10-27kg. Upright subscripts Sloping subscripts In other parts of ISO 31, the special names for the de- Cg (g gas) Cp (p: pressure) rived CGS units are given in informative annexes which are gn (n: normal) ∑nanδn (n: running num- not integral parts of the standards. ber) There are certain units outside the SI which are recog- µr (r: relative) ∑xaxbx (x: running num- nized by the CIPM as having to be retained for use together ber) with the SI, e.g. minute, hour and electronvolt. These units Ek (k: kinetic) gik (i, k: running num- are given in Tables 5 and 6. bers) Other coherent systems of units have been defined, e.g. χe (e: electric) px (x:x-coordinate) a system based on the units foot, pound and second and a T1/2 (1/2: half) lλ (λ wavelength) system based on the units metre, kilogram-force and sec- ond. NOTES Apart from these, other units have been defined which 8 Numbers as subscripts should be printed in roman (up- do not belong to any coherent system, e.g. the atmosphere, right type. However, letter symbols representing numbers the nautical mile and the curie. are generally printed in italic (sloping) type.

3 Recommendations for printing symbols and num- bers 3.1.3 Combination of symbols for quantities; elemen- 3.1 Symbols for quantities tary Operations with quantities

3.1.1 Symbols When symbols for quantities are combined in a prod- The symbols for physical quantities are generally sin- uct, this process of combination may be indicated in one gle letters of the Latin or Greek alphabet, sometimes with of the following ways: subscripts or other modifying signs. These symbols are printed in italic (sloping) type (irrespective of the type ab, a b, a·• b, a×b used in the rest of the text). The symbol is not followed by a full stop except for NOTES normal punctuation, e.g. at the end of a sentence. 10 In some fields, e.g. in vector analysis, distinction is NOTES made between a·• b and a × b.

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11 For multiplication of numbers, see 3.3.3. When a compound unit is formed by dividing one unit 12 In systems with limited character sets a dot on the line by another, this should be indicated in one of the follow- may be used instead of a half-high dot. ing ways: m Division of one quantity by another may be indicated m/s m∙s-1 in one of the following ways: s a A solidus (/) shall not be followed by a mul- , a/b or by writing the product of a and b-1, tiplication sign or a division sign an the same b line unless parentheses are inserted to avoid any e.g. a·∙ b-1 ambiguity. In complicated cases negative pow- ers or parentheses shall be used.

3.2 Names and symbols for units 3.2.3 Printing of symbols for units 3.2.1 International symbols for units No recommendation is made or implied When international symbols for units exist, they, and about the font of upright type in which symbols no other, shall be used. They shall be printed in roman (upright) type (irrespective of the type used in the rest of for units are to be printed. the text), shall remain unaltered in the plural shall be writ- ten without a final full stop (period) except for normal NOTE 15 In this series of publications the font used in punctuation, e.g. at the end of a sentence. such cases is generally that of the associated text, but this Any attachment to a unit symbol as a means of giving does not constitute a recommendation. information about the special nature of the quantity or context of measurement under consideration is incorrect. 3.2.4 Printing and use of prefixes Symbols for prefixes should be printed in roman (up- EXAMPLE right) type without a space between the symbol for the prefix and the symbol for the unit. Compound prefixes shall not be used. Umax = 500 V (not U = 500 Vmax)

The unit symbols shall in general be printed in lower EXAMPLE case letters except that the first letter is printed in upper -9 case when the name of the unit is derived from a proper Write nm (nanometre) for 10 m, not mμm. name. The symbol of a prefix is considered to be combined EXAMPLES with the single unit symbol to which it is directly attached, m metre forming with it a new symbol (for a decimal multiple or s second sub-multiple) which can be raised to a positive or negative A ampere power, and which can be combined with other unit sym- Wb weber bols to form symbols for compound units (see 3.2.2).

EXAMPLES 3.2.2 Combination of symbols for units 3 -2 3 -6 3 When a compound unit is formed by multiplication of 1cm = (10 m) =10 m 1 μs-1= (10-6 s)-1 = 106 s-1 two or more units, this should be indicated in one of the 3 3 following ways: 1 kA/m = (10 A)/m = 10 A/m N∙m, N m NOTE 16 For historical reasons the name of the base unit NOTES or mass, the kilogram, contains the name of the SI prefix ,"kilo". Names of the decimal multiples and sub-multiples 13 In systems with limited character sets a dot on the line of the unit of mass are formed by adding the prefixes to may be used instead of a half high dot. the word ,,gram“, e.g. milligram (mg) instead of microkil- ogram (μkg). 14 The latter form may also be written without a space, provided that special care is taken when the symbol for 3.3 Numbers one of the units is the same as the symbol for a prefix. 3.3.1 Printing of numbers Numbers should generally be printed in roman (up- EXAMPLE right) type. mN means millinewton, not metre newton. ITTC Symbols 5 Principles of Notation 5.1 Excerpt of ISO 31 Version 2021 100

To facilitate the reading of numbers with many digits, these may be separated into suita- 3.5 Symbols for chemical elements and nuclides Symbols for chemical elements shall be written in ro- ble groups, preferably of three, counting from man (upright) type (irrespective of the type used in the rest the decimal sign towards the left and the right; of the text). The symbol is not followed by a full stop ex- the groups should be separated by a small space, cept for normal punctuation, e.g. at the end of a sentence. and never by a comma or a point, or by any other means. EXAMPLES

3.3.2Decimal sign H He C Ca The decimal sign is a comma on the line. If the magnitude of the number is less than unity, the decimal sign should be preceded by a zero. A complete list of the symbols for the chemical ele- ments is given in ISO 31-8:1992, annex A, and 150 31- NOTE 17 In documents in the English language. a dot is 9:1992, annex A. often used instead of a comma. If a dot is used, it should The attached subscripts or superscripts specifying a be on the line. In accordance with an ISO Council deci- nuclide or molecule shall have the following meanings sion, the decimal sign is a comma in ISO documents. and positions. The nucleon number (mass number) of a nuclide is shown in the left superscript position, e.g. 3.3.3 Multiplication of numbers 14 The sign for multiplication of numbers is a cross (×) N or a dot half-high (ꞏ). The number of atoms of a nuclide in a molecule is shown in the right subscript position, e.g. NOTES 14 18 If a dot half-high is used as the multiplication sign, a N2 comma should be used as the decimal sign. If a dot is used as the decimal sign, a cross should be used as the multi- The proton number (atomic number) may be indi- plication sign. cated in the left subscript position, e.g. 19. In ISO documents, the dot is not used directly between 64Gd numbers to indicate multiplication. 3.4 Expressions for quantities If necessary, a state of ionization or an excited state may be indicated in the right superscript position. The symbol of the unit shall be placed after the numerical value in the expression for a quan- EXAMPLES tity, leaving a space between the numerical value and the unit symbol. It should be noted State of ionization: Na+ 3 3- that, in accordance with this rule, the symbol °C P O4 or (PO4) for degree Celsius shall be preceded by a space Electronic excited state: He*•, N0*• when expressing a Celsius temperature. Nuclear excited state: 110Ag*•, 110Agm The only exceptions to this rule are for the units de- gree, minute and second for plane angle, in which case there shall be no space between the numerical value and 3.6 Mathematical signs and symbols the unit symbol. If the quantity to be expressed is a sum or a difference Mathematical signs and symbols recommended for of quantities then either parentheses shall be used to com- use in the physical sciences and technology are given in bine the numerical values, placing the common unit sym- 1S031-11. bol after the complete numerical value, or the expression shall be written as the sum or difference of expressions for the quantities.

EXAMPLES l = 12 m - 7 m= (12 - 7) m = 5m t = 28.4 °C ± 0,2 °C = (28,4 ± 0,2) °C (not 28,4 ± 0,2 °C) λ.=220 × (1 ± 0,02) W/(m∙K)

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3.7 Greek alphabet (upright and sloping types)

alpha Α a Α a beta Β β Β β gamma Γ γ Γ γ delta Δ δ Δ δ epsilon Ε ε Ε ε zeta Ζ ζ Ζ ζ eta Η η Η η

theta Θ θ,  Θ θ, 

iota Ι ι Ι ι kappa Κ κ Κ κ lambda Λ λ Λ λ mu Μ μ Μ μ nu Ν ν Ν ν xi Ξ ξ Ξ ξ omicron Ο ο Ο ο pi Π π Π π rho P ρ Ρ ρ, sigma Σ σ Σ σ tau Τ τ Τ τ upsilon Υ υ Υ υ phi Φ φ Φ φ chi Χ χ Χ χ psi Ψ ψ Ψ ψ omega Ω ω Ω ω

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5.2 Computer Symbols 3. The Froude 'circular' symbols are defined by the Wherever possible the symbols in the second prefix CIRC. 4. All symbols start with a letter. column of the tables have been chosen so that 5. Qualifiers and operators, preferably two charac- their meaning is readily apparent. They have ters, are currently suffixed to the main symbol been constructed from the CCITT International line, without spacing. Telegraph Alphabet, restricted character set. 6. No one computer compatible symbol should be They are therefore suitable for use in a wide used for different concepts in a given context. This goal has not been completely achieved for range of situations e. g.: Telex messages, letters, the whole list. Ad hoc solutions have been at- computer printouts etc. tempted but discarded as unsatisfactory. To ensure that the symbols can be used in a 7. Since the computer compatible symbols have been wide range of programming languages they cur- proposed as the basis of attribute names for data ex- rently have been kept to less than six characters changes, the above rules will probably be further de- long. The symbols should be used as defined, veloped in the near future. A final remark on the Computer Symbols: in and, in accordance with modern programming the computer, the letter O and figure 0 (zero) practice, should have their type explicitly de- have fundamentally different meanings, but ow- clared before use. The following rules were ap- ing to their resemblance they can be easily con- plied in the derivation of the symbols: fused. Thus it is necessary to distinguish rigor- 1. Only upper case letter A - Z and digits 0 - 9 have ously between them. As a matter of fact there are been used. contradictory conventions being widely used. 2. Formerly Greek letters have been spelled out, if necessary in abbreviated form or with changed spelling. This practice is considered obsolete.

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5.3 Documentation 4. Japanese Translation of ITTC Standard Sym- 5.3.1 ITTC Documents bols. Transactions of the Society of Naval 1. International Towing Tank Conference, Architects of Japan, No.538, April 1974. Standard Symbols 1971, BSRA Technical 5. Russian Translation of ITTC Standard Sym- Memorandum No.400, August 1971. bols 1971. Brodarski Institute Publication 2. International Towing Tank Conference, No.28, Zagreb 1974. Standard Symbols 1976. BSRA T.M. 6. Simbolos Internacionales en Arquitectura Na- No.500, 1976. val. Asociacion de Investigacion de la Cons- 3. ITTC Dictionary of Ship Hydrodynamics. truccion Naval, RINA Maritime Technology Monograph Publication 7/75, Juli 1975, Madrid. No.6, 1978. 7. Report of Information Committee, Proc. 17th 4. Translation of Overall Index of Titles of Dic- ITTC, Göteborg 1984. tionary of Ship Hydrodynamics., Vol. 1: 8. Chinese Translation of ITTC Standard Sym- CETENA, Genova, 1984, bols. China Ship Scientific Research Centre, Vol. 2: University of Tokyo, 1984. Wuxi. 5. Bibliography and Proposed Symbols on Hy- drodynamic Technology as Related Model Tests of High Speed Marine Vehicles. 5.3.3 Other References Prep. by 17th ITTC High-Speed Marine Ve- Apart from the organizations represented on hicle Committee. SPPA Maritime Research the ITTC these symbols have been recom- and Consulting. Rep. No.101, 1984. mended for use in technical writing on naval ar- chitecture by a number of organizations con- cerned with marine matters including The Royal 5.3.2 Translations Institution of Naval Architects, the American A number of translations of the List of ITTC Society of Naval Architects and Marine Engi- Standard Symbols into languages other than neers and the American, British, Canadian, Aus- English have been made including French, Ger- tralian, and Italian Navies. Where possible, the man, Italian, Japanese, Russian, Spanish and symbols for Section 3.4.1, Waves are consistent Chinese. For obvious reasons these translations with the IAHR/PIANC List of Sea State Param- are no longer up-to-date as the present accepted eters, Supplement to Bulletin No 52, January list in English and the Russian one. 1986. 1. French Translation of ITTC Standard Sym- In 1985 the Draught International Standard bols 1971., Association Francaise de Nor- ISO/DIS 7463 Shipbuilding - Symbols for Com- malisation (AFNOR). puter Applications - has been published. The 2. International vereinbarte Buchstabensymbole symbols are based on the list approved by the und Bezeichnungen auf dem Gebiet der ITTC in Ottawa 1975 and a related list produced Schiffshydrodynamik. Collatz, G. by the ISSC in 1974, inconsistencies having Schiff und Hafen 27 (1975) No.10. been removed. The ISO/TC8/SC15 has been no- 3. Italian Translation of ITTC Standard Symbols tified that major changes of the ITTC Symbols 1971. Luise E. Appendix II, Report of are under discussion. Subsequently processing Presentation Committee. of ISO/DIS 7463 has not been postponed, but Proceedings 14th ITTC, Vol. 4, Ottawa 1975. the standard has been published as ISO 7463 in 1990.