Short gravity waves due to a steadily-advancing ship hull

Short gravity waves due to a steadily-advancmg ship hull

^ francis.noblesse@navy.mU; David Taylor Model Basin. NSWCXD, West Bethesda, MD, USA ^ gera1d.delho01meau@ec-n3ntes.fir; École Centrale de Nantes, C:NRS. Naates, France

* ffiuang@gmu.edu; Dépt of Computational and Data Scùnçes, Qœrgs Mason Universi^, Fairfax, VA, USA * cyang@gmû.edû; Dept of Computational and Data Sciences, Cîeorge Mason Univaraity, Fairfax, VA, USA

Iiitroduction

We consider potential flow about a ship bull thïü: advances at conistant speed along a straight path in calm water, using a Green function ttut satisfies the radiation cpndition and the Kelvin-K^cheU linear free-suif^e boundary condition. This approach e^qpresses the flow velocity u(x) generated by the ship Ki^l ^ a flow field point x ^ the S;Um of a local flow component and a wave conqœnent ûw (x) that is defined via a linear superposition of elementary water waves:

( I )

The wave-spectrum fimction S(t ; x) in this Füurier-Kbcbin representation of thé wave compcHiient uw is given by a

distribution of elementary w a ^ waves over the sMp hull. E.g., if the flow is r^iresented in terms of a distribution of sources, with drâsiQr <t, over thé surface H of the ship hull, we have

S{t;x) = ^ J d a ( x ) c ^ ^ - ^ ' - J * / ' — ' V i + ' - t - + * » ) / ^ V ( x ) (2)

Here,.ife stands for the 'e&ctive* portion of tfae bull surface H located ahead of the plane a; = x , in accradance with the step funqtipn H{x—x) in the expressirai for the Green function, and above the boräontal plane z = -7F^/{l+t^ ) because die ncponential fiinction in die integrand of (2) is smaller than 0.1% below diis plane.

The X axis is talcen along the patii of tfae ship and points toward die ship bow, tbe Z axis is vertical and points upward, and the mean free suface is taken as the p ^ e Z = 0. The flow is observed from a moving system of Cartesian coordinates {X, Y, Z) attached lo the ship and thus appears steady. The flow yelpcisy in tids moving frame of reference is given by{U-V>,M Wy^hsxe K stands for the ship speed and Ù =(U, V,W) is the (disturbance) flow due to flie ship. The coordinates of ^ flow-field point x in (1) and the source poim x in (2) are nondimensional in tenus of die ship length La, and F = K /VgLa is die Froude nûmbà.

The waterBne integral in tbe Neuoiami-Kelyin tfaeory of ship waves has a large effect

Before we consider sh(»t waves, we note that the classical Neumann-Kdvin thet^ of ^ p waves involves both a surface distribution of sources over die mean ship hull H, considered in (2X and a line di^ribution of scnnces around tbe mean ship wat»;line T. Specific^y, die Neumann-Kelvin theory of ship waves expresses the flow velocity u as

(3) whräe flie unit •rector n = ( n " , « " , « * ) is normal to the ship hull surface H and pomts into the water. The line distribution of sourms around tbe watärline f has ä large influence, as shown in Fig.1 fen* the particular case in whidb the source density er in (3) teken as o- = in accordance wiüi the sleoder-ship approximation proposed in [1].

Spedfically, Fig. 1 shows wave profiles for the usual parabolic Wigley huU (with parabolic frametines) at six Froude numbers F = 0.25 , 0.267 , 0.289. 0J16 . 0.354 , 0.408. The symbols in this figure conrespond to experimental m|»suraments performed at flie Umyasity of T0I90. 1 ^ two tfaeoretica] waye profiles in Fig.\ conespond to the slender-ship qrproximation defined by (3), widi 7 = n'', and flie related a^^iroximation in which the watCTline integral around T is i^uued. IM£Gn:»ices between these two slcndcr-ship approximations are significant, even dioug^ the line integral is 0(n^ )^ and the huU-surface integral is 0(n*}. In particular, die slen<^-ship ^proximation û ' ' is significantiy less osdUatory than tiu related slender-ship ï^roximatión û ^ . In fliis respect, the i^>proxiniatipn

+ is in better agreeiiKnt with experânental measurements than the approximation S^. However, die bow wave predicted by the approximation is significantly higher, ahd closo* to experim^t^ tneasurements. than flie bow waye ^ven by u^-H u^. The large nitiuence of the waterime integral in (3) is easily explained dieotetically [1].

There is a practical need for filtering short gravity waves

Tte wavelength A s 27rF^/(l + t^)<tf the elementary water waves in the Fourier integral (I) vanishes as t -+ ± 0 0 . Short gravity waves are not physically realistic because they are significantiy aSected by surface tension and viscosity, ignored in (1). Fuithmhore, short waves haye littie practical effect on main flow characteristics relevant to determine

Delft UhiVersHy of Technology

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Mekelweg 2 26282 CD Delft

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F «0.267 F « 0 . 2 « •0.1». F > a 2 n 4 J 44 4J-0.2 4.1 D 0.1 Oa lU M 01 0. 4.« 4 J ' 4 4 .0.3 .02 4.1 0 0.1 93 O.S 04 0.1 0.B 4 « ' 4 . > 4 4 « 1 - O J A I D 0.1 OJ 0.9 04 OJ OJ P - 0 . 4 M F-D.316 4 J -04 -03 47 4.1 0 D.1 U 04 OJ O 40 « S 44 4 J 4 J 4,1 0 0.1 0.3 03 04'OJ OO 4 J ' 4 J 44 4 J '«a'4.1 0 0.1 03 03 04 03 03

Figure 1: Waveprofilesgîvraby tbe slends-ship approximation (3), with a = n=^, and the related slendcr-sfaip app?oxim^on ü ^ , in which tiie waterline integral around F is ignored, f ^ die parabolic Wigley hnll àt six Rt>iide h u m b ^ F. The symbols represent experimental measurements.

the wave profile alôo^ the ship waterline, the velocity and pressure distribudons at ùne ship hull, and the related wave drag and hydrodynamic lift and momenL Lastiy. a huge number of very smaU panels would be reqmred to evaluate short waves witùn the frarnewo^ of a low-order panel method, fiideed, flie practirâl need fat filtering unrealistic md trotiblesoiiie short gravity waves is well understood, and the filtering of ätoci waves is an important aspect of a numerical evaluation of the flow velocity defined by a flow representation diat ignores viscosi^ and surface t«ision.

Short gravity waves can be filtered in a simple way

Waves widi waveleogdis A shorter dian some cncofi' wavelength A M are then eliminated here. The choice of flie cutoff wavelength Aoo is a matter of judgmem, to some extent. An obvipus dioice is to take Ac» as a small firaction of the wavelengdi 2xF^ of flie transveise waves genenùed by a ship advancing at a Froude number F. However, diis cdioice is meaningfid only at low Fronde mmibers. bideed, at h i j ^ Froude nsmbens, die

to the transverse dimensions — notably the beam B and flie draft D — of fl» ship tuül, which largely detennine die yarüttion of the flow velod^. Ttuss, at high Iroiide numbers, it is reasonable to choose die cutoff wavelength Ac» asa small ficaction of a (nondioiensiDnal) diazacteristic transverse dimension L°°/La of the ship hull. The choice

yields Z ^ ^ = Ä for a diiphuU widi circular cross section of radius R, fss D as B/,2if D fa B / 2 , L ^ ° as2D

if D « B and Z,*^as B if B < Z). We also haveX^^ < B and < 2Z>. The ratio

<7^ =

•BD

T - T - — ; Wltii b= -=r- and

d=-=-6/2 + d La La

(4) is a nondimoisional transverse shq) dimension m 'htdl-slendemess' parameter. For instance, we have er" = 0.06 for &=: 0.15 and d = 0.06.

n« cutoff wavelraigth Abo is dien taken as Aoo = eoSiri^'* or A « , - Soofr^ as F 0 ôt F oo, respectively. The relation A « , =

2xi?7(

1+<i

j

1 2 5

i o o « - 7 = f o r F < f f and t o o « ^ forFt<F widi Ft=OA.I~V<T'

ff"

(5) Thus, t o o isproportional toFexcqïtatlowFroüdenumbe^ F < j F t . Forso = 0.03, we have « 5.8 for F < Ft. wifliFt SB 0.13 for Ê O 0 =0.05 and a*'=0.06. These values of e«, and er" a l s o - y i d d t o o « 45.6F for Ft < F,i..e.

Hltering öf short waves is mösüy required if 0. Indeed, the exponential fimction E = e*^"'"*'*'^^' vanishes (lïçidly) a s t - + o o i f 2 < 0,butF — 1 for all values óf t if 2 0. Convergence of the Fourier integral (1) therefore is sigiiificanfly slower for z = 0 Äan forz< 0. The function E is flien multiplied by a function A that vanishes as £ -> OO ; e.g.,

A ^ e - ^ ' * l ' ' / * ~ . (6a) The fimction (6a) differs from 1 by less flian 1% for 0 < |*| < t. s CT.*«, with CT* » 0.3, 0.5,0.7. 0.9 for AT = 5,

9, 17, 59. The Fourier refoesentatión (1) of the velocity components uw, vw and ww involve the factracs i, it or

Vl +• . respectively. The functions t and V T + É ^ evidenfly incarease ïüce t:as i -> 00. Thus, convergence of die

Fburi^ i n t ^ a l (1) is significanfly slovver f w the components vw and ww dian for the compoitent ûw • It is then cbnveiiient to resplace t in the factors i t or y/l + t^ assocKUed wifli flie velocity components W and ww by *ö wifli

the function 0 defined as

ö = ( l + 1 0 0 | t | ' ^ - V i i ^ ) / ( l + 100|i|'^/i^^). (6b) We have td tast 0,t$ last 00, mdtO = tccBoo = { t o o +100)/101 for t = t o o . We flien have

1 < t o o Ö o o < 1.2for 1 < too < 20. Tlœ fimction (6b) diffère from I by less flian l%for0 < | t | < t . = i r . t o o wifli

CT, fa Q.3, 0.5, 0.7, 0.9 for Af = 8, 13, 25, 87. Tlie approximations A « 1 and sa 1 for 0 < | t | < i . mean fliat no a|)preciable wave damping ocau:s widnn die range 0 < 111 <tm and diat filtmng mosdy occurs within the range

tm < | t | < too • "Hiis wave-dampii^ r a i ^ represents 10% of the integration range 0 < | t | < too for CT. = 0.9, which

thuis corresponds to a fairly fast and computationally efficient fil;er. Hdwevo*, for CT, = 0.5, the wave-damping range represents 50% of ibe integration range, which means a slower and less efficient filter. The moderately efBcient filter (7. — 0.7, i.e. N = 17and Af = 25, is a reasonable comimimise. Thus, the Fourier representation (1) is modified as

I \=:^Qmr''dtAy/U^e^'-''"^'^^^^^^ ite \sit;x ) (6c)

where A arid 9 are givdi by (6a) with iV = 17 and (6b) with Af = 25.

Galccdations, fora modified V^gley hull diat has triaoigular (instead of pandwlic) fiamelines, of the wave con^Kment ù w defin«i by (1) and (2) with Ifae source density CT taken as CT = n'^. show tfaat both the fimctions A and ^ haye significant ejects, and tha£ large râlues óf the param^eis N and M are preföable. When used in addition to the filter A , die fiber B only has a relatively small benefidal effect on v and w, as expected because the function A already filters short waves. Hie two filta^ A and 9 can thai be used together, e.g. wifli die consistent values = 17 and Af = 25 that both correspond tp CT, fa 0.7. Calculattons show fl^ JV = 17 and Af = 25 indeâi are reasonable choices, and diat the dioices eoo =0.05ai^co = 0.03 in expressions (5) for ifae upper limit of integration too are reasonable.

Thc left, center and r i ^ a>lumns in Vi%2 show die wave conqKinents u i v , and ww * respectively, ^ven by (5) with JV = 17, Af = 25 and (2) with CT = n"^ for a modified Wigley hull (with triang^dar firameUnes) st five Frou^ mimbers F = 0.15 (t(^ row), 0.25, 0.35, 0.5 and 1 (bottom row). The low-speed filtering parameter EQ in (5) is cho»n as £0 = 0.03 for the calculations reported in Fig.1. where three values eoo = 0.01, 0.05. 0.1 of flie high-q[)eed filtering pmameter £ 0 0 û e considered. For die modified W ^ l ^ hull considered here, for which vve have 6 = 1/10, d = 1/16 ffluj CT^= 1/18, die transition Ftoude numb^ Ft defined by (5) is qiproximately equal to O.ÔS, 0.12.0.17 for £ œ - 0.01.0.05, 0.1 and £0 — 0.03. These Froude numbers are below flte range 0.15 < F < 1 consido^ in Fig.2, except in one case that corresponds to F = 0.15 and £ 0 0 = 0.1. H ü dunces £ 0 0 = 0.01,0.05, 0.1 yield vetocUies ûw. vw. ww that are not very d i f f e r s A relatively large value of £ 0 0 can then be chosen, a choice diat lowers die u{q>er limit of integration too defined by (5) and tfaerefore reduces computational efforts. Fig.2 also shows thi^ differences arc more irnportant for ânaller Froude nûmbràs, for which short gravity waves appear tp haw relatively more notable e&cts. The transverse comportents vw and (especially) ww of the flow velocity nw in ng.2 are ai^ireciably l a r ^ than the longitudind component ûw. in acccnxlance vndi tfae dieory of flow about a slender body in translatory motitm.

Cohcliision

The !^cHt-waye filter defined by (6) widi (S) and (4) is defined expUcidy, and a priori, in terms of die speed and the major dimensions Oeogfli, beain and draft) Pf a ship. Hie classical Neumann-Kelvin theoiy of sbip waves involves a line integral around die ship vralo'line diat has a large influence. Further details may be found in [2], where a practical mathonatical rqiresentatibn of the flow representation (3) is given.

References

[1] Noblesse F (1983) A slender-shq) dieory of wave resistance, J. S\àp Research, 27:1>33

[2] Noblesse F, Ddhommeau G, Huang F, Vang C (stdnnitted) Practical madiematical representation of die flow due to a distribution of soiuces on a steadily-advancmg ship hull

Acknowlei^inents

P B O . I B P-0.15 F<«.tS e,-o.as

*r-*—

*—*

_{i i ! r ^}

Hgme 2: Wave components Ûw Çeft). vvr (center), wtv (right) given by (5) witii JV = 17, A f = 25 and (2) vvitii = rt' for ä modified Wgley hull at F = 0.15 (top), 0.25, 035,0.5,1 (bottom). The tiuee curves correspond to = 0.01 0.05.0.1 with eo = 0.03.