Cover Image

PAPERBACK
$203.25



View/Hide Left Panel
Click for next page ( 454


The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement



Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 453
A Mode] for the Generation and Evolution of an ~ner-Angle Soliton In a Kelvin Wake R. Hall, S. Buchsbaum (Science Applications International Corporation, USA) ABSTRACT We develop a simple model for the generation and evolution of an inner- angle soliton in a Kelvin wake. The generation is modeled as an interference maximum due to a source-sink pair and the evolution is modelled using the nonlinear Schrodinger equation. The model is used to explain the results of a recent experiment. Although some of the parameters in the model are fit to experimental data, the values of the parameters are physically reasonable, thus we conclude that the model captures the essential physics. 1. INTRODUCTION It has long been observed that rays appear inside the cusp line within the diverging portion of the Kelvin wake of a ship. The simplest explanation for these rays is an interference pattern due to the superposition of the wave fields generated by the bow and the stern. Interest in these rays has increased in recent years because they are a possible explanation for some of the long bright lines observed in imagery of ship wakes collected from space [1]. Since these rays can form as a result of an interference pattern, they will appear in any model for the Kelvin wake of a ship as long at that model is capable of calculating the far-field wake. If the wave amplitude in the ray is small, or if the Kelvin wake model uses a linearized form of the free surface 453 boundary condition, then each ray inside the cusp line diverges linearly as it propagates aft: the width increases linearly with distance aft and the amplitude decreases as the inverse of the square root of the distance aft (the cusp line, of course, diverges slower). In a recent experiment [2,3], a ray in the Kelvin wake of a Coast Guard cutter was studied. Figures 1 and 2 show the near- and far-field wakes, respectively. The ship speed was 15 knots (the Froude number was 0.5) and the ray appeared at an angle of 10.9 degrees from -the wake cerIt-erline, which is close to the angle predicted by the interference model discussed above. However, the far-field evolution of the ray was not consistent with linear theory. Although Figure 2 shows that the width of the ray increased somewhat over the first few ship lengths aft of the stern, by a few hundred meters aft the width did not increase and the ray was shown to be an oblique nonlinear solitary wave packet. Related theoretical work [4] on the nonlinear evolution of a wave packet showed similar behavior. In this paper we will develop a simple physical model for the generation and evolution of the inner-angle ray observed in the experiment. In Section 2, the generation of the ray will be modeled using a simple linear theory in which the bow of the cutter is represented by a source and the stern is represented by a sink. In Section 3, the evolution of the ray into an oblique nonlinear solitary wave packet will be modeled using a two

OCR for page 453
:::::::::::::: I: ::::::::::::::: : ~ Ad :: :: ::: - Figure 1. The near-field wake of the Coast Guard Cutter Point grower. The ship speed is 15 knots. The lower photo shows the port side cusp line and soliton. The soliton intersects the lower left hand corner of the photo. 454

OCR for page 453
455 . ~ ~ ._, . - U, ~ a: o ~ , ~ o a, ~ A= .= ~ AD o a; , ~ .~ :: Cal Ct o a' o By Ct Ct 3 ~ . 1 Ct Cal Cal _. 0 Ct a' ~9 - Cal ~ C) ~ o : o ~ - _ o Cal - _. 4 - o o ~ o Cal Cal - _( 4_ o Cal o 4 - .0 o en Cal o Ct Cal 4 - o o AD o Cal U. a' C) 3

OCR for page 453
dimensional nonlinear Schrodinger equation for the complex packet envelope. Compared to the current state- of-the-art in modeling the Kelvin wakes of ships, our model is crude, but it has the virtue of capturing the essential physics in a simple way. In the conclusions we will review the limitations of the model and will suggest future improvements. Included in the Appendix is brief review of Kelvin wake kinematics, including the geometry of the interference rays generated by a source- sink pair, and a list of the parameters of the soliton observed in the experiment. 2. LINEAR THEORY Our model for the generation of the ray consists of a source-sink pair in a potential flow with linearized free surface boundary conditions. The interior continuity equation is V2~=SB +Ss, the linearized dynamic boundary conditions on z=0 are A, -Unix = -P- 877 (2) and P=O, (3) and the linearized kinematic boundary condition on z=0 is ql - U0x = Oz (4) The coordinate system is fixed to the ship, which is moving with speed ~ i n the positive x direction. y is positive to port and z is positive upward. ~ is the velocity potential for the perturbation to the uniform oncoming flow, ~ is the free surface elevation relative to the mean ambient level z=0, and P is the surface pressure divided by the density. The source at the bow, SB, and the sink at the stern, Ss, are given by point singularities: SB = UA (X - a)~(y) 8(z - c) ~ 5 ) and SS=-SB, with a - -a. (6) The depth of the source is -c and the distance between the source and the sink is 2a . Note that x = 0 is amidships; hereinafter, when we refer to distance aft, we mean distance aft of amidships, or -x. The volume flux emitted by the source is UA. In the limit that a))~, A is the cross-sectional area amidships of the region occupied by the fluid emitted from the source. Since the source is steady and the coordinate system is fixed to the ship, the solution of Equations (1~- (6) will be steady. In the experiment discussed in [ 1], the ship was a Point Class Coast Guard cutter (Point grower) and the ship speed was 15 knots. We obtain the required parameters for our model from Coast Guard Drawing No. 82(D) WPB- 0700-1, "82 Foot Patrol Boat (Class D) Docking Plan." The forward and aft (at the stern) perpendiculars are defined using a draft of 5'-3" above the base of the propeller. The distance between them is 78' (23.8 meters), which is the waterline length for this draft. The cross-section nearest amidships shown in the plan is a bulkhead located 36' aft of the forward perpendicular. Using the 5'-3" waterline, this cross-section has a beam of 4.4 meters at the waterline, a draft of 1.0 meters above the bottom of the bare hull, and a cross-sectional area of 3.3 square meters below the waterline. We do not know the actual static waterline or the running sinkage and trim for the experiment. For our model, we will use U = ~ 5 knots and we will let A be 3.3 square meters, which is the submerged cross- sectional area at the 36' bulkhead. The depth of the source, -c, will be half the draft at the bulkhead, or 0.5 meters. We could let the distance between the source 456

OCR for page 453
and the sink, 2a, be the 23.8 meter waterline length, but we chose a smaller value of 22.2 meters instead. This value is chosen so that the position of the ray in the model is fit to the observed position of the soliton in the experiment' as shown in the Appendix. It is reasonable that the value of 2a be less than the waterline length because the wave generation regions lie between the forward and aft perpendiculars. The solution of Equations ( 1)-(6) is given in terms of a superposition of two of the Green functions defined in Equation (7) of [5]. Figure 3a shows the surface elevation out to a distance of 400 meters aft. Only the far field portion of the solution, the portion given by Equation (7b) of [5l, is shown. The first ray inside the cusp line is the ray that evolves into the soliton. The ray appears at an angle of 10.9 degrees from the wake centerline and the wave vector at the peak of the ray is (kx, k,)=(0.426, l.Ol)m~i (see the Appendix). Unfortunately, there are two problems with this solution. First, the wave slopes within the diverging portion of the Kelvin wake are unrealistically large. This can be seen more clearly in Figure 4, which shows the cross-track slope ,7, at 100 meters aft. Second, two additional rays, which were not observed in the experiment, appear inside the first ray. Both of these problems can be corrected by replacing the point sources at the bow and stern with distributed sources. This is a reasonable modification of the model because the bow and stern regions are of finite extent. For convenience, we chose a Gaussian distribution in the horizontal plane, SB = 2 exp{- ( 2 ~ 2 Y |~(z - c) ~ 7 ) and SS=-SB, with a - -a, (8) because it reduces each spectral component in the solution by the simple factor exp(-~2k2 /2), where k is the horizontal wavenumber. Figure 3b shows the surface elevation for the solution of Equations ( 1~-~4), (7) and (8), with a value of 1.2 meters for cr. This particular value is chosen so that the amplitude of the soliton obtained in Section 2 agrees with the observations. The value is reasonable because it yields a value of 2.8 meters for the full width at half height of the Gaussian, which is consistent with the length scales in the bow and stern. Figure 4 shows the cross- track slope at 100 meters aft for this solution. Note that the values are now physically reasonable. 1.5 LLI 1 53 0.5 ~ O Jim -0.5 1 -1.5 | POINT | | DISTRIBUTED | X=-100M ~ ~ ~ I ~ I ~ 0 5 1 0 1 5 20 25 30 35 40 Y(M) Figure 4. A comparison of the cross- track slope at 1 00m aft for the two types of sources. 3. NONLINEAR THEORY We model the evolution of the ray into an oblique nonlinear solitary wave packet by using a time-dependent, two- dimensional nonlinear Schrodinger equation for the complex packet envelope. One procedure for doing this is to take the cross-track profile of the ray at a couple of ship lengths aft and 457

OCR for page 453
fir OSI SZI OOI SL (YV) A OSI SZI OOl SL (lo) OS SZ O 458 JO O ~ 4 - . - _4 o He V o an Z Ct O _4 o .0 Ct ~4 'Cal ._ Cal o o O ._, Cal ~ a, ') O C) ~4 0 O ~ V O ~ C ~ . - :: "D O ~ lo, ~ In ~ Cd Cal :e .= ~ a' O C) , ~ ~ Cal ~4 ._. 0 a, 4:, ~ ~ 3 o o o ~ ~ ~: cd - ~ ~ c - , CQ ~ O - 3 Cd a, 4 - Ct o . ~ C) I_, . ~ Ct _' o 1 ~ Ct au

OCR for page 453
use it as an inflow boundary condition for the nonlinear Schrodinger equation. A second procedure, which is suitable when the generation model is linear, is to bandpass the source term for the narrow band of spectral components that generate the ray, and then include this narrow-band source term in the nonlinear Schrodinger equation. We adopt this second procedure. We begin by returning to the generation model and separating the velocity potential ~ into a double body term (DB and a wave term 4' = (DB + . (10) The double-body flow satisfies the interior equation V2$DB =SB +SS (1 1) and the bondary conditions {DBZ = 0 and PDB = UDBX on z=O. The wave field satisfies the interior equation v2 = o and the boundary conditions ,-Ux =PDB grl (15) and 1, - UNIX = {z (16) on z=O. We solve Equations (11~-(13) for PDB and bandpass the result so that it only contains the narrow band of spectral components PDB that generate the ray. The rigorous procedure for bandpassing PDB involves finding the wavevectors of the two nodes in the Kelvin wake that bound the ray, then using a top hat filter to pass all of the spectral components along the Kelvin wake dispersion curve in between these nodes. For simplicity, we adopt an alternate approximate procedure: we replace the top hat filter with a Gaussian filter, we approximate the spectrum of PDB in the vicinity of the filter by a uniform value for the source and another uniform for the sink, And we set each uniform value equal the value at the spectral component corresponding to the peak of the ray. The resulting expression for PDB is PDB = Re tH exp[i (ken + kyy)~l, ( 17) where H=HB+HS' (12) HE=- k UAexp[- 2 -kc-i (kxa)] FB, ( 13) and (18) (19) Hs=-HB, with a - -a, (20) (14) and where FB is given by CThCTt,, ~ CT,` 2(X-a)2+~t,,2y2 1 B 2~ ~ ~ (21) (kx, ky)=(0.426, l.Ol)m~t is the wavevector of the Kelvin wave at the peak of the ray, and k = l.lOm-~ and ~ = 3.23rad / s are the wavenumber and frequency, respectively (see the Appendix). We choose a value of ~, = k' /2 for the standard deviation of the Gaussian filter in the ky direction. This is the most important direction because it is roughly parallel to the 459

OCR for page 453
Kelvin wake dispersion curve in the _ iamb I',12 .~2 (kX kit) plane. This value of a~ does a reasonable job of passing the spectral components that lie between the two nodes that bound the ray, as well as excluding the spectral components beyond the nodes (see the Appendix). The value of a~ must be large enough to pass the spectral components that lie between the nodes. We let it equal cry, for convenience. Although this is a few times larger than necessary, the additional filter width is normal to the Kelvin wake dispersion curve and does not contribute to the far-field wake. As suming that the surface 0.15 elevation has the narrow-band form O 0.05-~ -0.05 LL -o. 1 5 ,7 = Re {BeXp[i (fix+ kyy)]~' (22) we use standard procedures (see [6]) on Equations ( 14~-~17) to obtain the linear Schrodinger equation for the complex packet envelope: B. - fJBx = - 28 (kxBx + kyBy ) + 8~ 3 t(2k~2~k~2)Bxx -6kyk'Bxu, + (2kx2 - k'2 )Bn +i~H 2g (23) The solution of this equation is shown in Figure 5a. We note that the ray in Figure 3b appears somewhat larger that the ray in Figure 5a only because the former ray is superimposed on the transverse wave (note the different gray scales in the figures). The actual wave amplitudes are comparable. The nonlinear Schrodinger equation is obtained by using the nonlinear versions of Equations (15) and (16) in the derivation of Equation (23) (see [6]~. The well-known result is to add the term 460 (24) to the right hand side of Equation (23~. This term represents the increase in the phase speed of the wave due to nonlinearity. The solution of Equation (23), with (24), is shown in Figure Sb. The nonlinear term arrests the dispersion of the ray yielding an oblique nonlinear solitary wave packet. Figures 6a and 6b compare the linear and nonlinear Schrodinger equation results at 1 OOm and 400m aft, respectively. X=-4noM . . . 50 60 70 80 90 100 1 10 Y(M) 0 25 0.15 - - 0 0.05 ~ -0.05 us X=-1 OOM -0 26- ~, , 0 1 0 20 30 40 50 60 Y(M) | NONLINEAR | | ~- LINEAR | Figures 6a (bottom) and 6b (top). A comparison of the solutions of the linear and nonlinear Schrodinger equations. We note that we have performed additional calculations with the model

OCR for page 453
OSI SZI OOI SL (IAI) OSI SZI OOI SL (1~1) OS SZ 461 o z o o ,`_ o - E- Ct _ ~ o ~: - h7 Ct ~~ ~. - o V z z o z Ct ~3 z ~ - C. C~ ~ o ~ Ct .~ ca ~ _) ._ ~ D _ ~ 4_ C~ U3 ~ ~ _ Cd pL, C~ cn 4 - c: C) O ,~ ._ ~ a., O I_, c~ ct ._ _) ~ o ,7 O ~ ~: . ~ C~ Ct S . ~ a' O C~ ~ _ ~4 Ct ~ o t o C~ :: O ,= - 4- - o ;> 4 - ~ ~ _ _ ~ ~ Ct _ ~ D _ ^ O _) r~ ~ - C _ , ._, C~ ao Ct a~ ~: _} Ct 4_ O ~ '4- _ Ct ~ ~ Ct

OCR for page 453
using reduced forcing. If the peak slope of the soliton just aft of the source-sink region drops to 0.1, then a soliton does not form by 400 m aft. The required conditions on the initial packet for soliton formation are discussed in [3]. 4. CONCLUSIONS We have developed a simple model for the generation and evolution of the inner-angle ray observed in the experiment. The generation is modeled as an interference maximum due to a source-sink pair and the evolution is modelled using the nonlinear Schrodinger equation. Although some of the parameters in the model are fit to experimental data, the values of the parameters are physically reasonable, thus we conclude that the model captures the essential physics. We have three recommendations for future research. First, the generation model should be improved using one of the state-of-the-art Kelvin wake models, with the objective of replacing our fit with a true prediction. This is particularly important at low Froude number, where there are a larger number of interference rays and a model with a detailed representaion of the hull form and the near-field flow is essential. Second, the output from the generation model should be used as an inflow condition for the far-field equations, rather than using the double-body pressure forcing, as in our evolution model. The double-body pressure forcing is only suitable when the generation model is linear. Third, the far-field model should be upgraded to include a higher order expansion [6] or a fully-nonlinear equation. This will improve the approximation of the linear dispersion relation near the nodes and will add a nonlinear correction to the group velocity. ACKNOWLEDGEMENTS We thank Carl Scragg for the use of some of his linear Kelvin wake codes. REFERENCES Munk, W.H., Scully-Power, P., and Zachariasen, F., " Ships from space, " Proceedings of the Royal Society of London, Series A, Vol. 412, 1987, pp. 23 1 -254. 2. Brown, E.D., Buchsbaum, S.B., Hall, R.E., Penhune, J.P., Schmitt, K.F., Watson, K.M., and Wyatt, D.C., "Observations of a nonlinear solitary wave packet in the Kelvin wake of a ship," Journal of Fluid Mechanics, Vol. 204, 1989, pp. 263-293. 3. Buchsbaum, S.B., "Ship Wakes and Solitons," Ph D Thesis, University of California, San Diego, 1990. 4. Akylas, T.R., Kung, T.-J., and Hall, R.E., "Nonlinear Groups in Ship Wakes," Proceedings, 1 7th ONR Symposium on Naval Hydrodynamics, The Hague, the Netherlands, National Academy Press, Washington, D.C., 1988. 5. Noblesse, F., "Alternative integral representations for the Green function of the theory of ship wave resistance," Journal of E n g i n e e r i n g Mathematics, Vol. 15, No. 4, 1981, pp. 24 1 - 265. 6. Dysthe, K.B., "Note on a modification to the nonlinear Schrodinger equation for application to deep water waves," [~2ceedings of the Royal Society of London, Series A, Vol. 369, 1979, pp. 105-1 14. APPENDIX In this appendix we review Kelvin wake geometry. The condition that a wave is steady with respect to the ship is k-=U, (Al) where the dispersion relation for the wave iS m-~>0 k_4kx2+k'2 >0. (A2) 462

OCR for page 453
The group velocity of the wave determines its position in the wake, given by the angle a with respect to the wake centerline: 8~/ Ok U - de i at (A3) The condition for an interference maximum in a source-sink pair model is 2akx ens, n odd. (A4) The condition for an interference minimum is Equation (A4) with n even. The observed frequency of the soliton in the experiment was 3.28 rad/sec [21. Using a ship speed of 15 knots (7.7 m/s), the above formula imply that (kx, k')=(0.426, l.Ol)m~~ and a =10.9 deg. Equation (A4) then yields a value of 2a=22.2m for the distance between the source and sink that yields a ray at the observed angle. Using this distance, Equations (A4) and (A3) yield a value of 15.9 degrees and 8.3 degrees for the nodes on either side of the ray peals. The k, wavenumbers of these nodes are 0.393 m~l and 1.854 ma, respectively. Based upon an average of 24 runs beyond 0.5 km aft [2], the spatial width of the feature was 8.9m (measured at 1/e of the peak) and the peak wave amplitude was 15.1 cm (this was 1.1 times the theoretical soliton value calculated from the other parameters). An average of the runs at 0.5 km aft yielded a peak amplitude of 20 cm. 463

OCR for page 453