Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994) / Chapter Skim
Currently Skimming:
Session 2- Wavy/Free Surface Flow: Panel Methods 2
Session 2- Wavy/Free Surface Flow: Panel Methods 2
Pages 43-92
The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.
From page 45... ...
N Ni p RW,L SB sl s2 SF1 SF so SOO U x,y,z Or P : wave resistance coefficient midship section coefficient wetted surface area coefficient flow domain inner region of the flow domain outer region of the flow domain Froude number gravitational acceleration Green's function ship length, beam, and draft, respectively intersection of ship hull and free surface path of the line integral unit outer normal vector total number of nodes shape function pressure wave resistance and lift on a body, respectively wetted surface of ship boundary of the inner region boundary of the outer region free surface of the inner region free surface of the outer region interface of the inner and outer regions boundary surface at infinity ship speed Cartesian coordinate system solid angle at a control point wave elevation total velocity potential disturbance potential wave disturbance potential double-body disturbance potential fluid density
|
From page 46... ...
In the inner region the effect of nonlinearity of the free surface is very important, so that the finite element method is used together with the simplified nonlinear free surface condition. The effect of nonlinearity is negligible in the outer region far from the ship hull, where the Green function method is employed with the linear free surface condition.
|
From page 47... ...
2.2 Simplification of the Free Surface Condition The ship wavemaking problem described by (2)
|
From page 48... ...
Since we assume that the ship speed is low, the small difference will lead to invalidation of the assumption, therefore in the numerical calculation we only keep groups A, B C and D in the free surface condition: 0¢p 202 ~20(Pr i9 ~ - F - - OF ma n And n ~,~ 0gc2 2 0({r 15 So 26S¢r -2Fn - - F By NIX By n ~,~ ~y2 4 tar 03(p 02
|
From page 49... ...
In the inner region around the ship hull, the finite element method is used to solve the nonlinear problem; whereas, the effect of nonlinearity is negligible in the outer region far from the ship hull, thus the Green function method is adopted to solve the linear problem. By matching the solutions of inner and outer regions at their interface, the solution for the entire flow field can be obtained.
|
From page 50... ...
d: (33) In the outer region D2, we have the linear free surface condition: (~ + Fn2 49 A = 0 After discretization, we can write the matrix form [P]
|
From page 51... ...
3.3 The Finite Element Method for ~ in D~ According to the Galerkin method, for every Ni, ~1 (~, y, z) has to satisfy J J / V2~ NidV = 0 (37)
|
From page 52... ...
By solving (51) we can obtain So1 for the in 4 Numerical Results The numerical calculations have been carried out for wave resistance of a 2-D submerged cylinder, a 3-D submerged sphere, a Wigly model and a Series 60 Block 60 model.
|
From page 53... ...
5 Discussion and Conclud ing Remarks In the present study, based on the low-speed ship theory, an investigation has been made into the free surface conditions. Through a new approach of the order analysis, a simplified nonlinear free surface condition, which is different from Baba's and Dawson's, has been obtained.
|
From page 54... ...
Before using the coupled element method to the ship wavemaking problem, a special investigation was carried out to find the effect of the element dimension on the numerical results. Wave resistance and lift coefficients of a submerged sphere have been computed for different values of maximum element dimension Lm/(U2/g)
|
From page 55... ...
Shen, H.T., and Farell, C., "Numerical Calculation of the Wave Ir~tegrals in the Linearized Theory of Was ter Waves," Journal of Ship Research, Vol.21, No.1, 1977, pp.
|
From page 56... ...
4 Wave Resistance and Lift Coefficients of a Submerged Sphere ~z,~linear f.
|
From page 57... ...
For ships with significant flare or a transom stern a strip of panels forming a 'wake' trailing the ship is introduced and Kutta-type conditions of smooth detachment of the steady and unsteady flow at the stern are enforced. In steady flow, computations are presented of the Kelvin wake of a transom stern ship and in time-harmonic flow of the hydrodynamic coefficients, heave and pitch motions and wave induced structural loads for the SLY, S-175 hulls and an IACC yacht in head, beam and quartering waves.
|
From page 58... ...
More details on the steady flow may be found in Nakos and Sclavounos (1993J where the con~putatio~t of the wave resistance and the Kelvin wave patterns trailing transom stern vessels are discussed. In the linearized ti~ne-harn~onic problem, the enforcement of the smooth detachment condition at the transom is similar and equally important for the convergence of the numerical solution, ill analogy to the unsteady Kutta condition of finite velocity in lifting flows.
|
From page 59... ...
The nonlinear free surface condition is linearized on the assumption that either the ship hull form is slender or that the ship speed is low. Both these linearization assumptions may be acco~nmodated by the double-body linearization of the free-surface condition, which assumes the decomposition of ~ into the double-body potential, I, the steady wave potential, ¢, and 59
|
From page 60... ...
. The linearized free surface conditions governing the steady and unsteady wave disturbances have been derived and discussed in Nakos and Sclavounos (1990~.
|
From page 61... ...
Bi-Quadratic Spline Scheme Stability Analysis A bi-quadratic spline basis function has been introduced for the approximation of the velocity potential over the ship hull and the free surface. The numerical properties of this representation for forward speed free surface flows were studied in Nakos and Sclavounos (1990bJ.
|
From page 62... ...
For values of the reduced frequency T = ~U/g less than 1/4, a component of the unsteady wave disturbance generated by the ship is known to propagate upstream. In such cases the same upstream radiation condition was found to generate convergent computations of the unsteady flow past the ship for Froude numbers typically larger than 0.15.
|
From page 63... ...
are expected to properly model stern flows past transom sterns of small or zero draft. In either case the two conditions are transferred on the z = 0 plane and enforced on the upstream side of the wake.
|
From page 64... ...
where ~ is the unsteady pressure, A,, is the steady and 17r the relative wave elevations along the ship waterline, the latter defined as the difference between the total unsteady wave elevation and the vertical ship displacement. The corresponding moment vector acting over the saline portion of the ship hull, about the axes of the coordinate system centered at the point 0, is defined as follows M /// (ad-~ ~ x 32Tdm // path-~0)
|
From page 65... ...
7. NUMERICAL RESULTS 7.1 Kelvin Wave Patterns Past Transom Stern Ships Figure 4 illustrates the hull discretization of the transoms stern DTMB model 5415 consisting of 80 panels along the ship length and 10 panels along its half section.
|
From page 66... ...
The wave induced loads have been determined using both the linear and nonlinear heave and pitch motion amplitudes evaluated in the manner described in Section 7 Furthermore the contribution to the loads front the waterline integrals in (6.3)
|
From page 67... ...
The corresponding head wave shear force and bending moment computations for the S-175 hull are presented in Figure 12. Again the wave load predictions based on the 'nonlinear' heave and pitch motions are in better agreement with the experimental measurements and as for the S-175 hull are found to differ appreciably from their linear counterparts.
|
From page 68... ...
II. Raven, H., 1992, "A Practical Nonlinear Method for Calculating Ship Wavemaking and Wave Resistance", Proceedings of the 19th Symposium on Naval Hydrodynamics, Seoul, Korea.
|
From page 69... ...
4290 Panels Figure 2: Hull and Free Surface Discretization for S-175 Hull.
|
From page 70... ...
= .. ~Ve ~ Figure 4: Wave Pattern and Spectrum for the Model 5415 at Fr = 0.25.
|
From page 71... ...
~ D ED ~1 1.00t r' Tc 0.1S 0.10 0.05 A~ 0.00 ED 4.05 0.10 -0.16 -020 ox 020 0.16 Gin 0.05 o.oo 0.10 oboe ~ D odor 0.04 : 0.02 _ Coot 1.76t 50L 12S . D~porimont SWAN ...~__....I....I....I....I....I ~ ~ - ~ A rim 1 · ~ ~ I ~ r ~ · ~ .
|
From page 72... ...
~1 f 1~W~ o ~o s~ '.°t -180 ~ 2~00 1 ~SOO ~D 1.~ 0.100 _ 0.050 ~ D o.ooo ~ns _ ~ D Ct . -0.10 _ -0.15 _ - A£,S H5S S 2.0 2.S 3.0 3S ~J /E 9 ~1 Elsporimorlt SWJ[N ~ 2.000 _ ~ D Q 1.000 _ .
|
From page 73... ...
0.00 _ _ _ I I I I _ ..... I I ~ I 0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 ~Jo \/: w0 ~ Figure 9: Vertical Shear Force and Bending Moment at NIidship Section of the SL-7 Hull in Head Waves at Fr = 0.3.
|
From page 74... ...
. 1 2 we ~ g Figure JO: Heave and Pitch Motions for the S-175 Hull in Head, Beam and Quartering Waves at Fr = 0.275.
|
From page 75... ...
. _ _ _ _ _ _ i / 1 / ~ 1 1 1 1, 1 __________ 1 · \ 1 1 1 1 1 __________ 1 1 1 1 1 1 1 1 _ _ _ _ _ _ _ _ _ _ 1 1 1 1 1 1 1 .S 2.0 2.S 2 S Body Plan, Discretization and Heave & Pitch Motions for PACT IACC Yacht Hull in Head Waves at Fr = 0.33.
|
From page 76... ...
/~" W~ / \ . ~ 1.0 2~0 ' ~3.0 4.0 I.0 .~0 wo;~9 Figure 12: Vertical Shear Force and Bending Moment at Midship Section of the S-175 Hull in Head, Beam and Quartering Waves at Fr = 0.275.
|
From page 77... ...
The amplitude of these effects is magnified in the mid-ship bending moment and shear force integrations which are carried out over half the ship surface. Similar interference effects in the exciting force evaluation are less pronounced due to cancellation in the respective integration which is carried out along the entire ship length.
|
From page 79... ...
The integral equation involves line integrals around the hull and has been designed so that it greatly reduces the requirement of using forward finite differences to enforce the radiation boundary condition. The approach permits the presence of free-surface dipoles and provides a mechanism for carrying lift downstream from a transom stern in accordance with the work of various researchers who, in recent years, had suggested that lift effects are an important component of wave resistance under certain conditions.
|
From page 80... ...
A Rankine singularity integral equation is presented for obtaining the solution of the resulting mathematical boundary value problem. The integral equation involves line integrals around the hull and has been designed so that it greatly reduces the requirement of using forward finite differences to enforce the radiation boundary condition.
|
From page 81... ...
It is intended to solve the boundary value problem with a low-order Rankine singularity panel code in which the source and dipole distributions over each panel are uniform. Such codes often use finite differences to approximate the derivative of the potential in the last term of the integral equation.
|
From page 82... ...
an r dS(~) -F2 56 2 J Or dS(: SF Transom Stern Flow past a ship with a transom steno is now considered.
|
From page 83... ...
1-F2 ~ Airs d71 (30) This equation is the final form of the integral equation for the case of a transom stern hull.
|
From page 84... ...
Since disturbances propagate downstream, this treatment of ¢~5E at the downstream boundary should not affect the flow near the ship if the downstream boundary is far enough downstream. Hull-waterline integrals are split into a part along the hull-transon~ intersection and a part along the sides of the hull.
|
From page 85... ...
transom. Instead, equations that would be associated with these collocation points are replaced with equations specifying free-surface depth and slope in terns of the hull geometry at the transom stern.
|
From page 86... ...
PREDICTIONS Athena Hull Plots of the computed Kelvin wave pattern and the amplitude of the free~wave spectrum obtained front the computed wave pattern are presented for speeds corresponding to the Froude numbers 0 48 and 0.4. The first speed is especially interesting because the wave elevation immediately aft of the transom steno can be compared with published measurements.
|
From page 87... ...
Solid lines indicate positive nondimensional wave elevation z/F2 = 0.005j forj - 1, 2, etc. Dashed lines indicate negative wave elevation z/F2 _ -0.005j for j - 1, 2, etc.
|
From page 88... ...
A comparison of the amplitude of the fiee-wave spectrum obtained front computed wave elevations along transverse wave cuts and from measured wave elevations is presented in Fig.
|
From page 89... ...
Solid lines indicate positive nondimensional wave elevation z/F2 = 0.005j ford = 1, 2, etc. Dashed lines indicate negative wave elevation Z/F2 = 0.005j forj = 1, 2, etc.
|
From page 90... ...
As a second refinement, the nonlinear zero-pressure Kutta condition, Bemoulli's equation with the pressure set to at~nospheric pressure, could be satisfied at the transom stern. This nonlinear Kutta condition should be used in conjunction with a varying depth in the surface originating at the hull-transom intersection about which the free-surface boundary conditions are ]
|
From page 91... ...
21 Amplitude of the free-wave spectrum obtained from a pair of computed transverse wave cuts at 1.15 and 1.20 ship lengths aft of the midship of Model 5415 Froude number 0.25 :3 and the corresponding spectrum obtained from experimental data Aid. Paneling aft of the transom stern has been placed on the mean free-surface level.
|
From page 92... ...
D "Kelvin Wakes and Wave Resistance of Cruiser and Transom Stern Ships," Journal of Ship Research, to appear.
|
Key Terms
This material may be derived from roughly machine-read images, and so is provided only to facilitate research.
More
information on Chapter Skim is available.