is the instantaneous position vector of any grid point or fluid

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1 Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in integal and diffeential foms in absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume and evaluate the advantages and disadvantages of using the thee coodinates. Appoach The govening diffeential equations of motion ae deived and solved in the absolute inetial eath-fixed coodinates (X,Y,Z), non-inetial ship-fixed coodinates (x,y,z), and elative inetial coodinates (X,Y,Z ) that tanslate at a constant velocity in (i.e. with espect to) (X,Y,Z) fo an abitay moving but non-defoming contol volume (C), as shown in Figue 1. The contol volume is fo eithe the ship in ed o backgound in geen while the solution domain includes both. The oigin of (x,y,z) o is located at the cente of gavity of the ship. The oigin of (X,Y,Z ) is located at the intesection of the bow and the static wate line. paticle in (x,y,z). is the instantaneous position vecto of any gid point o fluid In the absolute inetial eath-fixed coodinates (Fig. 1(a)), the position vecto of o is. S = + is the instantaneous position vectos of any gid point o fluid paticle. The ship tanslation velocity is = t = ui ˆ ˆ i + vi j + wk i, whee ˆ î, ĵ, and ˆk ae the unit vectos of X, Y, and Z axes, espectively, and u i, v i, and w i ae the suge, sway, and heave velocities Ω=Ω ˆ +Ω ˆ +Ω ˆ of of the ship espect to (X,Y,Z). The ship otation is angula velocity Xi Y j Zk (x,y,z) in (X,Y,Z). The velocity of the contol volume is defined by: = +Ω (1) 1

2 In elative inetial coodinates (Fig. 1(b)), the position vecto of o is in (X,Y,Z ). S S = + S = + and ae the instantaneous position vectos of any gid point o fluid paticle in (X,Y,Z ) and (X,Y,Z), espectively. The ship tanslation velocity is = t = ui ˆ + v ˆj + wk ˆ i i i (X,Y,Z). u i, v i, and w i, whee unit vectos of (X,Y,Z ) ae the same as those fo ae the suge, sway, and heave velocities of the ship espect to (X,Y,Z ). The ship otation is angula velocity Ω=Ω X i +Ω Y j +Ω Z k of (x,y,z) in (X,Y,Z ). The velocities of the contol volume in (X,Y,Z ) and (X,Y,Z) ae: = +Ω (2) = + +Ω whee the elative inetial coodinates ae moving at a constant speed C espect to (X,Y,Z), = iˆ C. Compaison between equations (2) and (3) shows: ˆ ˆ ˆ (3) = + (4) The ed contol volume pefoms up to 6 degees of feedom (6DF) motions (suge, sway, heave, oll, pitch, and yaw). The geen contol volume pefoms up to 3DF (suge, sway, yaw) motions copied fom the coesponding degees of feedom of the ship s motions. Any numbe of degees of feedom can be imposed and the est is pedicted by 6DF solves, which esults in captive, fee, o semi-captive motions. î, ĵ, and ˆk ae the unit vectos of x, y, and z axes, espectively. 2

3 (a) (b) Figue 1 Definition of diffeent coodinates fo ship Hydodynamics simulations with absolute inetial eath-fixed coodinates (X,Y,Z), non-inetial ship-fixed coodinates (x,y,z), and elative inetial coodinates (X,Y,Z ): (a) solve in (X,Y,Z), (b) solve in (X,Y,Z ). 1. elocity coelations 1.1 Coelation between (X,Y,Z) and (x,y,z) If epesents the position vecto of a fluid paticle in (x,y,z), the position vecto of that fluid paticle in (X,Y,Z) is: S = + (5) The absolute velocity in (X,Y,Z) and elative velocity in (x,y,z) ae defined by Eqns. (6) and (7), espectively: = ds (6) = d (7) Diffeentiate Eqn. (5) is diffeentiated espect to time and with use of the coelation of d in (X,Y,Z) and d in (x,y,z) [eenwood, 1988]: 3

4 The velocities ae elated by: d d = +Ω (8) = + (9) 1.2 Coelation between (X,Y,Z) and (X,Y,Z ) Deivation simila to 1.1 is applied in (X,Y,Z ). The absolute velocity in (X,Y,Z) and elative velocity in (X,Y,Z ) ae defined by equations (1) and (11), espectively: Use Equation (8) and diffeentiate S = + espect to time: = ds (1) = ds (11) = + (12) Diffeentiate S S = + espect to time: = d + S = + (13) 2. eynolds tanspot theoem (TT) TT fo flow with an abitay C moving at db SYS d = + is applied: Fo a non-defoming C, TT fo incompessible flow: db syst βρ d βρ n da (14) C β = ρ + ρ β d nda (15) C To convet it to the diffeential fom of TT, the flux tem is tansfomed to a volume 4

5 integal using the auss divegence theoem: db syst β = ρ d + ρ ( β ) d C C C β = ρ + ( β ) d (16) Taking limit fo elemental C fo which integand is independent of the volume and divided by : db syst β = ρ + ( β ) (17) 3. Application of TT fo ( XYZ,, ): 3.1 Integal fom fo a C Consevation of mass with = M, β = 1 fo Eqn.(15) and note steady flow with =, Consevation of momentum with ( ) nda= (18) = M and β = fo Eqn. (15): dm Fo steady flow: 3.2 Diffeential fom = F = ρ d + ρ ( ) nda (19) C F = ρ ( ) nda (2) Fo mass consevation, substitute β = and = (when the = M, 1 5

6 C is limited to a point, is the local gid velocity so = ) to Eqn. (17) and note = : = (21) Fo momentum equation, substitute = M and β = to Eqn. (17): d M = syst + = f ρ (22) The 2 nd tem on the left-hand-side can be simplified using the divegence opeato expansion with the use of the continuity equation, = + (23) = continuity Afte the body and suface foces ae expessed pe unit volume, the NS equation in (X,Y,Z) is: ρ + ( ) = ( p+ γz ) + μ Application of TT fo ( X, Y, Z ) 4.1 Integal fom fo a C Fo mass consevation, steady flow: (24) 2 : = M and 1 β = fo Eqn. (15) and note =. Fo nda= (25) 6

7 Consevation of momentum with B M ( ) syst = +, β = + and = ( + ) dm [ ] = ( + ) F = ρ d + ρ ( + ) nda C (26) Fo steady flow with the use of continuity: F = ρ + nda = ρ nda+ ρ nda (27) F = ρ nda (28) 4.2 Diffeential fom Fo mass consevation, eplace in Eqn. (21) using Eqn. (13): Fo consevation of momentum, = (29) and in equation (24) ae substituted by Eqns. (4) and (13). Note the time deivatives in (X,Y,Z ) and (X,Y,Z) ae the same as they ae both inetial coodinates, tem [1] in equation (24) becomes: Fo tem [2] in Eqn. (24), = + = ( ) ( ) = + + = (3) (31) 7

8 Fo tem [3] in Eqn. (24), = + = (32) Equations (3) to (32) ae substituted back to equation (24) and note gadient, divegence, and Laplacian opeatos ae fame invaiant. The govening equations in tems of ae obtained: ( 2 ) p Z ρ + = + γ + μ (33) 5. Application of TT fo (x,y,z): 5.1 Integal fom fo a C Fo mass consevation, = M, β = 1 fo Eqn.(15) and note steady flow, nda= (34) Consevation of momentum with B M M ( ) syst = = +, β = d M = F = ρ d + ρ ( + ) nda C = ρ d + ρ nda + ρ nda C (35) ( ) Ω = + + = + + 2Ω +Ω +Ω ( Ω ) (36) a el 8

9 So, whee Ω Ω Ω Ω= = +Ω Ω= F a dm = ρ d + ρ nda = el C C (37) (38) Note when the C is moving togethe with (x,y,z), the will be eplaced by Fin the above equations 5.2 Diffeential fom Fo mass consevation, eplace in Eqn. (21) using Eqn. (9): = (39) The diffeential fom fo momentum equation in (x,y,z) can be deived fom Eqn. (24), note = +Ω (4) = + (41) Fo tem [1], fom Eqn.(36), Fo tem [2], = + a el (42) = + = (43) 9

10 Fo tem [3], = + = (44) The above fou equations ae substituted back to the diffeential fom of the momentum equation in (X,Y,Z) and note gadient, divegence, and Laplacian opeatos ae fame invaiant. The govening equations in tems of whee ae obtained: 2 ρ + = ρael p+ γz + μ body foce (45) a = + 2Ω +Ω Ω +Ω el (46) 6. Comments on application of diffeent appoaches 6.1 Solving fluid mechanics poblems analytically using integal foms As shown in the lectue notes, it is easie to solve the continuity and momentum equation in the elative inetial coodinates. 6.2 Solving diffeential equations using CFD [Xing et al., 28] The govening diffeential equations (DEs) fo continuity and momentum in (X,Y,Z) and (X,Y,Z ) ae tansfomed fom the physical domain in Catesian coodinates ( X, Y, Z, t ) to the computational domain in non-othogonal cuvilinea coodinates ( ξ, η, ζ, τ ) using the chain ule without involving gid velocity fo the time deivative tansfomation: 1 J j ξ j ( bu i i) = (47) 1

11 j k k l Ui 1 k Ui 1 k p 1 bi bi U b i j ν U t b i j + b ( U U ) = b + S k k j k + + k l τ J ξ J ξ J ξ J eeff ξ J ξ J ξ j j j i i (48) Eqs. (47) and (48) ae identical to Wasi et al., who tansfomed the standad NS equations in ( X, Y, Z, t) to a fixed computational domain ( ξ, η, ζ, τ ) with the inclusion of the gid velocity X i τ in the time deivative tansfomation. In the pesent deivation of Eqs. (24) and (33) the solution domain and contol volume ae coincident with each othe, which conceptually bette shows the elationship between the moving but non-defoming contol volume and solution domain and additionally povides the continuous / fom of the NS equations. Compaed with Eq. (33), Eq. (24) allows non-constant C. When C =, Eq. (33) educes to Eq. (24). Tansfomation of the continuous fom of the NS equations in (X,Y,Z) into (x,y,z) clealy shows the diffeence of DEs using the absolute inetial eath-fixed o non-inetial ship-fixed coodinate systems. Compaed with Eq. (45), application of Eq. (24) simplifies the specification of bounday conditions, saves computational cost by educing the solution domain size, and can be easily applied to simulate multi-objects such as ship-ship inteactions. In geneal, implementation of Eq. (24) to simulate captive, semi-captive, o full 6DF ship motions is staightfowad. efeences [1] Xing, T., Caica, P., Sten, F., Computational Towing Tank Pocedues fo Single un Cuves of esistance and Populsion, ASME Jounal of Fluids Engineeing, ol. 13, No.1, 28, pp [2] eenwood, D. T., 1988, Pinciples of Dynamics, 2nd edition, Pentice-Hall Inc [3] Wasi, Z. U. A., 25, Fluid Dynamics, Theoetical and Computational Appoaches, 3d edition, Taylo & Fancis oup. 11