Theory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018

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1 Quiz No. 2, April 25, 2018 (1) viscous effects (2) shear stress (3) normal pressure (4) pursue (5) bear in mind (6) be denoted by (7) variation (8) adjacent surfaces (9) be composed of (10) integrand (11) be referred to as (12) partial differential equation (13) hold for (14) incompressible fluid (15) continuity equation (16) substantial derivative Let us consider a general volume integral of the form I(t) = F (x, t) dv. The time derivative of I(t), which is known as the transport theorem, can be written as di dt = V (t) where U n denotes the normal velocity of the boundary surface S. By using this transport theorem and Gauss theorem together with U n = u j n j ( 3 j=1 u j n j ), the principles of conservation of mass and momemtum expressed by d ρ dv = 0 dt can be transformed as follows: d ρu i dv = dt V (t) V (t) S(t) pn i ds + ρg δ i3 dv V (t)

2 Quiz No. 3, May 9 (Wednesday), 2018 (1) vorticity (2) scalar function (3) identity (4) be recast (5) atmospheric pressure (6) readily (7) normal vector (8) imply (9) kinematic condition (10) dynamic condition (11) wave elevation (12) eliminate (13) explicitly (14) expedient (15) pragmatic (16) subsequently Let us consider an ideal fluid with irrotational motion; that is, ω = u = 0 (where ω is the vorticity). In the vector analysis, it is known that an identity of Φ 0 is always satisfied irrespective of the kind of scalar function Φ. Then, write the relation in this case between u and Φ. Write the continuity equation (derived from the conservation of mass for incompressible fluid) in terms of u and also in terms of Φ. Euler s equations, which are to be obtained from the conservation of momentum, can be written in the form u t + u u = 1 p + gk (1) ρ where ρ denotes the density of fluid, p the pressure, g the acceleration due to gravity, and k the unit vector in the vertical z-axis. Then by using a transformation of the vector quantity u u = 1 2 ( u u ) in the case of u = 0, write the result of Eq. (1) in terms of Φ, which is known as Bernoulli s pressure equation. Write the substantial derivative of function F (x, y, z, t) = z ζ(x, y, t) = 0 being equal to zero, in terms of the velocity potential Φ instead of the velocity vector u.

3 Quiz No. 4, May 16, 2018 Let us consider a solution of the velocity potential for plane progressive waves, satisfying the following Laplace equation and the free-surface and deep-water conditions: [L] [F ] [B] 2 ϕ x ϕ = 0 for y 0 (1) y2 2 ϕ t 2 g ϕ = 0 y on y = 0 (2) ϕ = 0 y on y = h (3) Skipping the details (I believe you understand), we can write a homogeneous solution satisfying (1) and (3), in the following form (with D as unknown): ϕ(x, y, t) = D cosh k 0(y h) cosh k 0 h sin(ωt k 0 x) (4) However, the free-surface condition (2) is not imposed yet. In fact, the so-called dispersion relation can be obtained from Eq. (2). Write explicitly the relation to be obtained from Eq. (2). (5) Next, obtain asymptotic forms of Eqs. (4) and (5) in the limit of h. Then, from the dispersion relation to be obtained from Eq. (5) for the case of h, obtain the relation between period (T ) and wavelength (λ), and evaluate the wave period for λ = 100 m and the wavelength for T = 10 s. (You may use 2π/g 0.8.) (1) nevertheless (2) advection (3) specify (4) physically relevant (5) trivial (6) necessary and sufficient (7) sinusoidal (8) eigen value (9) homogeneous (10) spatial part (11) time-dependent (12) likewise (13) mutually (14) wave elevation (15) envisage (16) monotonically (17) estimation (18) schematically (19) no longer (20) provided that

4 Quiz No. 5, May 23, 2018 (1) Prove (confirm) Eq. (2.16) by substituting Eq. (2.15) in Eq. (2.10). (2) Determine two unknowns C and D in Eqs. (2.16) and (2.17) from the conditions of Eqs. (2.13) and (2.14), and show that the result can be expressed in a unified form of Eq. (2.18).

5 Quiz No. 6, May 30, 2018 Let us consider the mass conservation in the 2D potential flow by using the s 1 s 2 coordinate system. The total net flux through the boundary surface may be given as dq 1 + dq 2 = ( ) ϕ δs 2 δs 1 + ( ) ϕ δs 1 δs 2 = 0 (1) s 1 s 1 s 2 s 2 In the polar coordinate system, the differential element can be given by δs 1 = δr, δs 2 = rδθ. Thus by substituting these relations into Eq. (1) and dividing the result by rδrδθ, obtain the Laplace equation expressed in the 2D polar coordinate system. Let us consider the following integral: F (x, y) = 0 1 k e ky cos kx dk = Re k eik(x+iy) dk Re 0 k eikz dk, z x + iy (2) First consider differentiation of Eq. (2) with respect to z and then perform integration of the result (df/dz) with respect to k. From the result to be obtained, prove the following expression: where r = x 2 + y 2 and c is an arbitrary constant. F (x, y) = log r + c (1) complementary (2) modulation (3) slowly varying (4) envelope (5) encompass (6) persist (7) preceding (8) retain (9) discard (10) crest (11) trough (12) transportation (13) decay (14) subsequent (15) simplicity (16) be replaced with

6 Quiz No. 7, June 6, 2018 (1) decay (2) diverge (3) noteworthy (4) discard (5) envisage (6) quadrant (7) transient (8) plausible (9) residue theorem (10) without recourse to With assumption of x > 0 and y > 0, let us consider the following complex integral in the complex plane J C e ζy+iζx ζ K where C denotes a certain round integration path. When considering the integration path shown on the right side, we note that J = 0 because there are no singularities inside of the round integration path. With this fact and by taking only the real part of the result, please show the following: L C C 0 dζ e ky cos kx k K dk = 0 k cos ky K sin ky k 2 + K 2 i1 O for x>0 k = K e kx dk π e Ky sin Kx 1 k

7 Quiz No. 8, June 20, 2018 (1) reciprocity (2) artificial (3) argument (4) be imposed (5) advantageous (6) in return for (7) explicitly (8) that is to say (9) homogeneous condition (10) be regarded as (11) asymptotic form (12) wave elevation (13) characteristics (14) body geometry With Green s theorem, we can obtain the following expression for the velocity potential at an arbitrary point P = (x, y) in the fluid region: { } ϕ(q) ϕ(p) = ϕ(q) G(P; Q) ds(q). (1) n Q n Q S H where S H denotes the wetted surface of a floating body. Using this expression, let us consider the asymptotic form of ϕ(p) for the case of x. Since P(x, y) is included only in the Green function G(P; Q), we have the following expression for x : G(P; Q) i e K(y+η) ik(x ξ) = i e Kη±iKξ e Ky ikx as x ±. (2) In terms of this result, please show the asymptotic form of the velocity potential valid for x. Furthermore, show that the wave at a distance from the body ( x ) is the outgoing progressive wave, and obtain the complex amplitude of the wave elevation from the relation ζ(x) = iω g ϕ(x, 0).

8 Quiz No. 9, June 27, 2018 (1) characteristics (2) distinction (3) wave absorption (4) perfect reflection (5) superposition (6) incident wave (7) calm water (8) schematic (9) displacement (10) time invariable The body boundary condition is given by where Φ = α(t) n (1) n Φ(x, t) = Re [ ϕ(x) e iωt ], x = (x, y), n = (n 1, n 2 ) α(t) = iξ 1 (t) + jξ 2 (t) + kξ 3 (t) x, ξ j (t) = Re [ X j e iωt ] Without the time-dependent part e iωt, rewrite Eq. (1) in a form of summation of three modes (sway, heave, and roll) of body motion; that is, confirm the following result: ϕ 3 n = iωx j n j (2) j=1 If the velocity potential is written in the form 3 ϕ(x) = ϕ 0 (x) + ϕ 4 (x) + ϕ j (x) = ga { φ0 (x) + φ 4 (x) } 3 + iωx j φ j (x) (3) iω j=1 what form of the boundary condition does each velocity potential φ j (x) (j = 1, 2, 3, 4) have to satisfy? j=1

9 Quiz No. 10, June 27, 2018 (1) specifically (2) implicitly (3) wetted surface (4) discard (5) reference value (6) be associated with (7) transpose (8) transfer function (9) be referred to as (10) advent (11) variance (12) couple of vertical forces When the velocity potential is given in the following form ϕ(x, y) = ga ( ) 3 φ0 + φ 4 + iωx j φ j, φ 0 = e Ky+iKx (incident wave) iω j=1 and the asymptotic form of the velocity potential due to body disturbance can be written as φ j (x, y) ih ± j (K) e Ky e ikx as x ±, write the wave elevation on the free surface ζ(x) valid at a distance from the body and the pressure in the fluid p(x, y) by using the following relation: ζ(x) = iω g ϕ(x, 0), p(x, y) = ρiωϕ(x, y) Then write the hydrodynamic force acting on the body in the i-th direction F i

10 Quiz No. 11, July 4, 2018 (1) restoring moment (2) buoyancy (3) reflection (4) transmission (5) deformation (6) subsquently (7) asymmetric (8) indicate (9) in contrast to (10) nevertheless (11) take into account (12) likewise The wave elevation at a distance from a body can be expressed as ζ(x) = a e ikx + iah ± 4 (K) e ikx K 3 X j ih ± j (K) e ikx as x ± where we note that the incident wave comes from the positive x-axis. Then, write the equations for the nondimensional (in terms of the amplitude of incident wave a ) complex amplitude of reflection and transmission waves: j=1 Transform the following integral by using the 2D Gauss theorem and write the result in terms of the centers of gravity (G) and buoyancy (B), and the metacenter (M): S G 3 = ρgx 3 S H x { n 2 x n 1 (y OG) } dl where n 1 = n x and n 2 = n y.

11 Quiz No. 12, July 11, 2018 (1) in what follows (2) interpret (3) asymptotic (4) be imparted to (5) be exerted by (6) superficially (7) remarkable (8) extension (9) specifically (10) owing to (11) irrespective of (12) be associated with (13) consequence (14) hold for For two different velocity potentials, ϕ and ψ, satisfying the same boundary conditions on S F and S B but not necessarily the same on S H and S ±, the Green s theorem gives the following equation: S H ( ϕ ψ n ψ ϕ ) dl = 1 [ ( ϕ ψ n 2K x ψ ϕ ) ] x=+ x y=0 x= Then let us consider a combination of ϕ = φ D (diffraction) and ψ = φ j (radiation in the j-th mode). The boundary conditions satisfied by these on S H can be written as φ D n = 0, φ j n = n j and at x ±, the following relations hold: φ D e ikx + R e ikx, φ j ih + j e ikx, at x = + φ D T e ikx, φ j ih j eikx, at x = In this case, obtain the relation to be derived from Eq.(1): (1)

12 Quiz No. 13, July 18, 2018 (1) decompose (2) namely (3) asymmetric (4) antisymmetric (5) split equally (6) independently (7) variation (8) inversely (9) resonance (10) moment of inertia (11) preceding (12) noteworthy (13) step by step (14) general-shaped The equation of harmonic heave motion with circular frequency ω can be written in the form [ C 22 ω 2( m + A 22 ) + iω B22 ] X 2 = E 2 (1) where A 22, B 22, and C 22 are the added mass, damping coefficient, restoring-force coefficient, respectively, and E 2 denotes the (complex) wave-exciting force. By introducing the following relations C 22 ω 2( m + A 22 ) ρω 2 E 2, B 22 = ρω H + 2 2, E 2 = ρgah + 2 (H +2 = ik Ā2 e iε2 ) prove that the complex amplitude of heave motion X 2 /a will be given analytically in the following form X 2 a = cos α H ikh + e iα H = cos α H e i(α H+ε 2 ) Ā 2 2 where, tan α H = E 2 / H (Note that at the resonance, E = 0 and thus α H = 0.) (2)

13 Quiz No. 14, July 18, 2018 For a symmetric body, the relation between the scattered wave H 4 ± in the diffraction problem and the radiated wave H + j (j = 1 3) in the radiation problem can be shown to be expressed as H 4 ± = Im( H 2 + ) i Re( H + ) j (j = 1 or 3) (1) H + 2 H + j From (3.18) in the lecture note, the radiation Kochin function H + j ratio Āj and the phase difference ε j in the form can be written with the wave amplitude H + j = i K Āj e iε j (j = 1, 2, 3), where K = ω2 g (2) Then, rewrite Eq. (1) in terms of ε j (in fact, Ā j will not appear in the result). According to Section 4.1, the reflection-wave and transmission-wave coefficients in the diffraction problem can be expressed as Prove these relations. R = ih + 4 = cos(ε 2 ε j ) e i(ε2+εj) (3) T = 1 + ih 4 = i sin(ε 2 ε j ) e i(ε 2+ε j ) (4)