Wave Drift Force in a Two-Layer Fluid of Finite Depth

Transcription

1 Wave Drift Force in a Two-Layer Fluid of Finite Deth Masashi Kashiwagi Research Institute for Alied Mechanics, Kyushu University, Jaan Abstract Based on the momentum and energy conservation rinciles, a comact calculation formula is analytically derived for the wave drift force on a -D body floating in a two-layer fluid of finite deth. In a two-layer fluid, two different wave modes (the surface-wave mode with longer wavelength and the internal-wave mode with shorter wavelength) exist not only in the incident wave but also in the body-scattered wave, and these wave characteristics are roerly incororated in the obtained formula. It is noted that, unlike the single-layer case, the wave drift force can be negative in the incident wave of surface-wave mode, if the transmitted wave with internal-wave mode is large. Numerical comutations are imlemented for a Lewis-form body by means of the boundary integral-equation method with Green s function for the twolayer fluid roblem. The effects of density ratio, interface osition, and body motions on the wave drift force are studied, and some imortant features are found for two-layer fluids. Key words: Two-layer fluid, wave drift force, surface-wave mode, internal-wave mode, finite water deth. Introduction ydrodynamic studies of a body floating in a two-layer fluid of finite deth have been conducted for the radiation and diffraction roblems which are of first order with resect to the incidentwave amlitude ([] [] and the references therein). owever, to the author s knowledge, no study has been made on the second-order wave drift force in a two-layer fluid. For the case of single-layer fluid, it is well known that the coefficient of reflection wave is directly connected with the wave drift force and its calculation formula is established by the far-field method based on the momentum and energy conservation rinciles. For a two-layer fluid, however, the analyses look comlicated even for first-order roblems. For examle, in the diffraction roblem, two different incident waves of surface-wave mode (with longer wavelength) and internal-wave mode (with shorter wavelength) must be considered searately for a rescribed frequency, and each incident wave will be scattered by a body into two different wave modes. Thus the energy of the incident wave may be transferred from one mode to the other. Furthermore, when the body is oscillating in resonse to the incident wave, the body motion may change the reflected and transmitted waves. For this comlicated wave field in a two-layer fluid, it is crucial to understand analytically what is the correct form of the calculation formula for the wave drift force, in what way two-layer effects are incororated in the formula, and what are distinctive differences from the single-layer case. In this article, after the definition and formulation of the roblem, the asymtotic exression of the velocity otential valid at a far field is obtained, in which the coefficients of reflected and transmitted waves are defined in terms of the Kochin functions for the radiation and diffraction roblems in a two-layer fluid. Then on the basis of the momentum and energy conservation rinciles, analytical integrations at the far field and some mathematical transformations are erformed to derive the desired calculation formula for the wave drift force. The key to success in this rocedure is to aly the orthogonality roerties to the eigenfunctions in the two-layer fluid of finite deth. Numerical comutations are erformed for a Lewis-form body, using the boundary integralequation method develoed by Ten and Kashiwagi []. The density ratio and the interface osition between the uer and lower layers are varied and those effects on the wave drift force

2 are studied. Furthermore, by showing the results for three cases where the body is comletely fixed, only the heave motion is free, and all modes of body motion are free in resonse to incident waves, the effects of body motions on the wave drift force are also discussed. Lastly, some findings from the theoretical and numerical studies in this article are summarized as conclusions.. Mathematical Formulation We consider a -D floating body of general shae in a two-layer fluid with finite deth. The body may intersect the interface between the uer and lower layers and is assumed to oscillate sinusoidally in resonse to an incident wave with circular frequency ω. Fig. shows the Cartesian coordinate system and notations used in the analyses below, with the origin on the undisturbed free surface and the z-axis ositive in the downward direction. The free surface, the interface, and the flat rigid bottom of the water are located at z =,z =, and z = h, resectively. Assuming both of the uer and lower fluids to be incomressible and inviscid with irrotational motion, the velocity otential is introduced and written in the form Φ (m) (x, z, t) =Re [ (x, z) e iωt ], m =, where (x, z) = ϕ (m) D ga = = iω φ(m) D (x, z)+ 3 j= iωx j Rj (x, z) ga iω ϕ(m) (x, z), () 3 (x, z) = D (x, z) K j= X j Rj (x, z), (3) A (x, z) =φ(m) (x, z)+φ(m) (x, z), (4) I S with K = ω /g, and g is the gravitational acceleration. ere the suerscrit (m) denotes the fluid layer, with m = and corresonding to the uer and lower layers, resectively. As described in Yeung and Nguyen [3], there can be two different wave modes in the incident wave in two-layer fluids for a rescribed frequency. Those modes are differentiated with subscrit (), and secifically = is referred to as the surfacewave mode and = as the internal-wave mode. A is () denotes the amlitude of incident wave at each mode. It is known that a simle relation holds at each mode on the amlitude ratio between the waves on the free surface and on the interface. owever, the ratio between the waves of surface-wave and internal-wave modes on the free surface or the interface is not known a riori. Therefore the diffraction roblem h h S C S C () S I S F S S I () S () O n S F S () S I Incident wave S C S I () S C x z=h S B z Fig.. Coordinate system and notations

3 must be solved for two different incident waves at a given frequency. In this article, A at each mode is defined in the theory as the incident wave on the free surface, whereas in numerical comutations a larger amlitude is adoted as A at each wave mode (i.e., A is the amlitude on the free surface and A is the amlitude on the interface). D denotes the diffraction otential, which includes the incident wave otential φ(m) I to be given as the inut (exlicit exressions of which will be shown below) and the scattering otential S. φ(m) Rj in (3) is the radiation otential with unit velocity in the j-th direction (j = for sway, j = for heave, and j = 3 for roll) and the amlitude of the j-th mode of motion, X j/a, must be obtained by solving the equations of motions of a body, for which hydrodynamic forces must be comuted with the solution of the boundary-value roblem. The governing equation for the velocity otentials is the -D Lalace equation + = (5) and the linearized boundary conditions to be satisfied on the free surface, the interface, and the rigid bottom of the lower layer are exressed as follows: φ + Kφ = onz =, (6) φ = φ() ( ) φ γ + Kφ = φ() on z =, (7) + Kφ () φ () = onz = h (= + h ). (8) ere, by linearity, in the above can be any of the velocity otentials aearing in (4), and γ = ρ /ρ is the density ratio, with ρ m being the density of the uer (m =) and lower (m = ) fluids. Since the incident wave is indeendent of the resence of a body, the velocity otential of the incident wave, I, can be obtained from (5) (8) and by secifying the amlitude of the incident wave on the free surface (z = ) or the interface (z = ). As shown in Fig., the incident wave is assumed to roagate from the ositive x-axis. Then the velocity otential of the incident wave is exressed in the form where I (x, z) =Z(m) (k ; z) e ikx (9) Z k chkz K shkz (k; z) = k Z () K chkh k shkh (k; z) = chk(z h) k shkh and the variable k in (9) and () is the wavenumber satisfying the disersion relation for a two-layer fluid given by () D(k) =K ( k shkh K chkh ) + ε ( K k ) shk shkh =. () For brevity, the hyerbolic functions of cosh(x) and sinh(x) have been written as ch(x) and sh(x), resectively, and ε = γ in (); these notations will be used throughout this article. To determine the other velocity otentials associated with the disturbance by a body, the boundary condition on the body surface must be imosed, which can be given in the form D n Rj n = ( =, ) = nj (j = 3) 3 on S (m)

4 where n j denotes the j-th comonent (n = n x, n = n z, n 3 = xn z zn x) of the normal vector, which is defined as ositive when directed into the fluid domain from boundaries (see Fig. ). The boundary-value roblems for the disturbance velocity otentials (excet for the incident-wave comonent) may be comleted by imosing the radiation condition of generated waves radiating from the body. 3. Numerical Solution Method The diffraction and radiation otentials formulated above are determined directly by the integral-equation method in terms of the Green function satisfying all homogeneous boundary conditions. This solution method can be alied to a general case where an arbitrary body intersects the interface between the uer and lower layers, and the derivation of the integral equation based on Green s theorem is shown in Ten and Kashiwagi [] and Kashiwagi et al. []. The results may be summarized in the form C(P) l (P) + n= = S (n) l (Q) G (m) n (P; Q) ds n Q I (P) ( l = D; =, ) n j(q)g (m) n (P; Q) ds ( l = Rj; j = 3) n= S (n) where P = (x, z) and Q = (ξ,ζ) denote the field and integration oints, resectively, located on the body surface and C(P) denotes the solid angle. G (m) n (P; Q) reresents the Green function, which has different forms deending on whether P and Q are in the uer or lower layer; details of which are shown in Ten and Kashiwagi []. The so-called constant-anel collocation method is adoted for solving (3); that is, the body surface of z>is divided into N segments and on each segment the unknown velocity otential is assumed to be constant. Then, considering N different oints for P(x, z), (3) can be recast in a linear system of simultaneous equations for N unknowns. In actual numerical comutations, some additional field oints are considered on both z = and z = inside the body to get rid of the irregular frequencies. The resultant over-constrained simultaneous equations are solved using the least-squares method. Once the velocity otentials on the body surface are determined, it is straightforward to comute the hydrodynamic forces that must be used in solving the motion equations of a body in each of the incident waves of surface-wave mode ( = ) and internal-wave mode ( = ). The calculation method for the motions of a body in waves is described in Kashiwagi et al. []. 4. Velocity Potentials at the Far Field The analyses necessary for obtaining asymtotic exressions of the velocity otentials at x ± may be found also in Ten and Kashiwagi [] and Kashiwagi et al. [], and the results are summarized as follows: where D Rj φ(m) i ± S (k) = ± Rj(k) = (3) I + i S(k ± q) Z (m) (k q; z) e ikqx, (4) q= Rj(k ± q) Z (m) (k q; z) e ikqx, (5) q= n= n= S (n) φ (n) S (n) D { φ (n) Rj n W n(k; ζ) e ±ikξ ds, (6) n D (k) φ(n) Rj n Wn(k; ζ) D (k) e ±ikξ ds, (7) 4

5 W (k; ζ) =γα(k)k shkh Z (k; ζ) W (k; ζ) =α(k)k shkh Z () (k; ζ) α(k) = D (k) =K(shkh + kh chkh Khshkh) (8) K K chk k shk, (9) +ε { k shk shkh +(K k )( chk shkh + h shk chkh ). () Eqs. (6) and (7) are the Kochin functions (comlex amlitude functions of the bodydisturbance waves) comuted from canonical velocity otentials in the diffraction and radiation roblems. In terms of these Kochin functions and the comlex amlitude of the j-th mode of motion, X j/a, to be obtained by solving the motion equations of a body in the incident wave of k -wave mode, the Kochin function reresenting the whole disturbance wave with wavenumber k q can be obtained by linear suerosition as follows: ± (k q) ± S(k q) K 3 j= X j A ± Rj(k q). With this definition and taking summation of both comonents of the surface-wave (q = ) and internal-wave (q = ) modes, the asymtotic exression of the velocity otential defined by (3) can be exressed in the following form: ϕ (m) (x, z) Z (m) (k ; z) e ikx + ϕ (m) (x, z) R q Z (m) (k q; z) e ikqx as x +, () q= T q Z (m) (k q; z) e ikqx as x, (3) q= where R q = i + (k q) (4) T q = δ q + i (k q) with δ q being Kroenecker s delta. R q and T q defined in (4) can be understood as the coefficients of reflected and transmitted waves, resectively, of the k q-wave mode when the incident wave is of the k -wave mode. 5. Momentum Conservation Princile Following Maruo [4], a calculation formula for the wave drift force in the horizontal direction can be derived on the basis of the momentum and energy conservation rinciles. Let us consider first the momentum conservation rincile in the x-axis in a two-layer fluid. With the same transformation as that for a single layer fluid, the following equation may be obtained as a basis: { ( ) Φ (m) Φ (m) (m) n x + ρ m n Un ds =, (5) where m= S (m) S = S + S C + S I + S F S () = S () + S() C + S() I + S B (6) { Φ (m) (m) = ρ m + t Φ(m) Φ (m) + (m) S, (7) [ Φ (m) ga =Re iω ϕ(m) (x, z) e ]. iωt (8) 5

6 ere the overbar in (5) means the time average over one eriod and U n in (5) reresents the normal velocity of the boundaries surrounding the fluid under consideration. (m) S in (7) denotes the static ressure indeendent of the disturbance velocity otential. As exlicitly written in (8), only the incident wave of the k -wave mode ( = or ) is considered here. As shown in Fig., the control surface S (m) C in the resent study is arallel to the z-axis and in the linear theory the free surface S F, the interface S (m) I, and the bottom of fluid S B are arallel to the x-axis; these are fixed in sace and thus U n = on these boundaries. On the other hand, the normal velocity of the body boundary must be equal to the normal velocity of the fluid and thus U n = Φ(m) on S (m) n. (9) Taking these into consideration and retaining only quadratic terms in the velocity otential, an exression for the wave drift force acting in the negative direction of the x-axis may be written as follows: F D (m) n x ds m= S (m) [ = h { Φ Φ ρ Φ Φ [ + h { Φ () ρ ρ S F Φ () Φ() Φ () Φ Φ dx + Φ S Φ ρ dx Φ S () Φ () ρ dx. (3) () I I ere the square brackets with suerscrit + and subscrit in (3) means the difference between the quantities in the brackets evaluated at x =+ and x =. The integrand in the integrals on S F and S I may be transformed in terms of the boundary conditions given by (6) and (7) as follows: on S F Φ on S I ρ Φ = ρ γ Φ Φ Φ = KΦ Φ ρ Φ () γ K Φ () = K {Φ Φ, (3) = ρ γ Φ Φ = γ ρ γk For taking time average, the following formula may be useful {γφ Φ () { Φ Φ. (3) Re [ Ae iωt ] Re [ Be iωt ] = Re[ AB ], (33) where A and B are comlex in general and the asterisk means the comlex conjugate. Substituting (3) and (3) in (3) and alying (33) with (8) gives the following result: FD F D ρga [ = h K + [ ϕ z= { ϕ ϕ + γ [ + γ ϕ γk z= h { ϕ () ϕ() (34) 6

7 This may be regarded as an extension of the exression for a single-layer fluid to the case of two-layer fluid, and in fact the last line in (34) can be understood as contributions from the square of wave height at the free surface (z = ) and the interface (z = ). owever, this form is not convenient for analytical integration with resect to z, because the derivatives with resect to z are included. Thus it is not straightforward to utilize the orthogonality roerties of the eigenfunctions for a two-layer fluid summarized in Aendix. To surmount this inconvenience, we consider further transformation for the integrals including the derivatives with resect to z using the Lalace equation and the boundary conditions on z =,z =, and z = h. Performing artial integrations and substituting (5) (8), the following result can be justified: I h ϕ ϕ + γ { ϕ = K + γ z= γk + h ϕ h ϕ ϕ + γ () ϕ ϕ () h z= ϕ () ϕ (). (35) Substituting this result in (34), we can see that the first line on the right-hand side of (35) cancels exactly the last terms in (34) to be evaluated at z = and z =. Therefore, as a final result convenient for analytical integrations with resect to z, the following exression can be obtained: [ h F D = γk { ϕ w(z) ϕ ϕ (36) where w(z) and ϕ are defined as { w(z) =γ, ϕ = ϕ for z, w(z) =, ϕ = ϕ () for z h. (37) 6. Energy Conservation Princile A relation to be obtained from the energy conservation rincile is usually used to derive a comact formula for the wave drift force and also to check the accuracy of comuted results. With the same notations as for (5) (8), a basis equation for the two-layer fluid may be given in the form m= S (m) { ρ m Φ (m) t ( Φ (m) n (m) Φ (m) + ρ m )U n ds =. (38) t With this equation, considering the normal velocity U n of the boundaries, the work done by a body onto the fluid may be evaluated as follows: W (m) U n ds S (m) m= = ρ [ h + ρ S F Φ t Φ ρ [ h () Φ Φ () t Φ Φ t dx Φ S Φ ρ t dx + Φ S () Φ () ρ dx. (39) () t I I 7

8 Substituting (8) and taking time average over one eriod using the formula (33), the following result can be obtained: W W ( ω ρga K [ h = Im +Im S F ) ϕ ϕ ϕ + γ h ϕ dx + γ Im S I () ϕ { ϕ () ϕ() ϕ() γ ϕ ϕ dx (4) Im in the above means that only the imaginary art is to be taken. Taking account of the boundary conditions on S F and S I as did in deriving (3) and (3), one can easily rove that the integrals on S F and S I have no contributions because the integrands are of real quantities. Therefore, with notations in (37), the result can be written in the form W = γ Im [ h w(z) ϕ ϕ ere it is noteworthy that the work done by a body must be zero when the body is fixed (as in the diffraction roblem) or freely oscillating in waves without external oscillation devices sulying the energy. 7. Wave Drift Force aving reared all necessary equations, let us derive the formula for the wave drift in a twolayer fluid. The asymtotic exressions of ϕ (m), given by () and (3), must be substituted in (36). To erform this rocedure in general, () for instance can be written as { ϕ (m) (x, z) =Z (m) (k ; z) e ikx + R e ikx + Z (m) (k q; z) R q e ikqx, (4) with convention of =/ q; that is, when = (the incident wave is of the surface-wave mode) q =, and when = (the incident wave is of the internal-wave mode) q =. It should also be noted that, owing to the orthogonality in Aendix, there is no need to consider cross terms between k -wave and k q-wave in evaluating the integrals with resect to z. Therefore, using (4), the integrand at x =+ in (36) may be written as ϕ (m) ϕ(m) ϕ (m) (4) = { Z (m) (k ; z) k (+ R ) + { Z (m) (k q; z) k q Rq. (43) ere the integrals with resect to z can be identified with the normalization integral, whose exlicit form is rovided in Aendix as follows: F(k) k γ h w(z) { Z(k; z) = K (K chkh k shkh) + kh k γk sh kh + ε [ h ( k γ k K + ) ] (K ) (K k ) chkh k shkh + γ sh kh K K. (44) With these results, the integral at x =+ in (36) takes the following form: { ( k + ) R F(k )+k Rq q F(kq) [ F D ]+ = K. (45) 8

9 In the same manner, the integral at x = can be analytically erformed. Namely, with convention of =/ q, (3) is written as and then it follows that ϕ(m) ϕ (m) ϕ (m) (x, z) =T Z (m) (k ; z) e ikx + T q Z (m) (k q; z) e ikqx (46) ϕ (m) = { Z (m) (k ; z) k T + { Z (m) (k q; z) k q Tq. (47) Therefore, in terms of (44), the result of the integral at x = takes the form { k T F(k)+k Tq q F(kq) [ F D ] = K. (48) The wave drift force must be given by the difference between (45) and (48). Therefore the result from (36) is exressed as F D = [ ( k + R ) ( T Rq F(k )+k q ) ] Tq F(k q). (49) K As the next ste, let us consider a relation to be obtained from the energy conservation rincile given by (4). With the same convention for ϕ (m) and the orthogonal roerties for the integrals with resect to z, the final result of (4) can be exressed as follows: { ( W = R ) ( T Rq F(k ) ) + Tq F(k q). (5) As noted at the end of the receding section, W = for the case where a body is fixed or freely oscillating in the incident wave. Therefore the energy conservation rincile takes the form ( R T ) F(k )=( Rq + Tq ) F(k q). (5) Substituting this in (49), the final form of the calculation formula for the wave drift force in a two-layer fluid can be given in the form [ F D = { k R k + k q F(k)+ Rq k k q + ] Tq F(k q). (5) K By considering the limiting case of γ, let us confirm the corresonding formula for a single-layer fluid. For γ =,k and the internal wave no longer exists, and hence =, F(k )=, k = k, K = k tanh kh. (53) In this limiting case, ε = in (44); thus the coefficient associated with the normalization integral can be transformed into J k K F(k) =+ h (K chk k shk) K sh kh =+ h K (K shkh k chkh) =+ kh shkh. (54) Therefore it follows from (5) that { F D = FD = R + kh, (55) ρga shkh with R being the coefficient of reflected wave in a single-layer fluid. This result is well known as a formula for the wave drift force in the water of finite deth. Another thing noteworthy for (5) is a ossibility of negative value of the wave drift force in a two-layer fluid. The wave drift force to be comuted from (5) is mostly ositive. In articular for the case of = (i.e., in the incident wave of internal-wave mode), the value of 9

10 (5) is definitely ositive because k >k. owever, for the case of =, the value of (5) can be negative if the value of T (the transmitted wave with wavenumber k in the incident wave of the surface-wave mode) is relatively large. When the energy is not conserved due to viscous effects such as the viscous daming in roll, relation (5) obtained from the energy conservation rincile can not be used. owever, even in this case, the momentum conservation rincile holds and thus the wave drift force can be comuted with (49). 8. Numerical Results and Discussions Numerical comutations were erformed for the Lewis-form body used in the revious study of the first-order radiation (Ten and Kashiwagi []) and diffraction (Kashiwagi et al. []) roblems. This Lewis-form body can be reresented by a conformal maing with two nondimensional arameters; those are the half-breadth to draft ratio = b/d =.833 and the sectional area ratio σ = A/Bd =.9 (in real dimensions, the breadth B =b =.mand the draft d =. m). Since this body is symmetrical with resect to x, only half of the body surface was discretized into 4 segments for all comutations in this article. With this number of segments, satisfaction of the energy-conservation rincile given by (5) was virtually erfect with the F D/.5ρ ga Surface-wave mode F D/.5ρ ga Internal-wave mode =. =.7 (. ) =.9 =.7 =. =.9 Fig.. on a Lewis-form body ( =.833, σ =.9) in two-layer fluids: Effect of the difference in the fluid density for the case where all body motions are fixed. F D/.5ρ ga Surface-wave mode F D/.5ρ ga Internal-wave mode =.9 (.. ) =. =.7 =.7 =. =.9 Fig. 3. on a Lewis-form body ( =.833, σ =.9) in two-layer fluids: Effect of the difference in the fluid density for the case where only the heave motion is free to oscillate.

11 F D/.5ρ ga Surface-wave mode F D/.5ρ ga Internal-wave mode =.7 (.. ) =. =.9 =. =.7 =. =.7 =.9 Fig. 4. on a Lewis-form body ( =.833, σ =.9) in two-layer fluids: Effect of the difference in the fluid density for the case where all body motions are free to oscillate. order of error being 4 for both cases of body motions fixed and free to oscillate in waves. 8.. Effects of the density ratio To see the effects of the density ratio on the second-order wave drift force, comutations were imlemented for the same arameters as those in the study of the first-order radiation and diffraction roblems; that is, γ =.,.9,.7, and. with the deths of the fluid layers fixed at =.d and h =.d. As γ, the fluid reduces to a single-layer fluid of h =.d. Conversely as γ, the lower fluid behaves more like a rigid block, and the results are exected to aroach those for a single-layer fluid with uer layer deth =.d. To illustrate this behavior, comutations were also carried out for single-layer fluids of h =.d and.d. Figure shows the nondimensional value of the wave drift force for the case where all body motions are fixed, in which the left-hand and right-hand sides are for the surface-wave and internal-wave modes, resectively, of the incident wave. It should be noted first that the wave drift force shown in Fig. is always ositive over the whole range of frequency, although theoretically there is a ossibility of negative value in the incident wave of surface-wave mode. The results for γ =.9 are close to those for a single-layer fluid of h =.d excet for very low frequencies, and the results for γ =.7 are also almost the same as those for γ =.9in the frequency range of >.4. On the other hand, for γ =., the results in the incident wave of surface-wave mode tend to aroach the results for a single-layer fluid of h =.d, and the nondimensional value of the drift force in the incident wave of internal-wave mode also becomes large. (Note that the amlitude A of the incident wave of internal-wave mode is taken as that on the interface.) Figure 3 shows comuted results for the same arameters but for the case where only the heave motion is free to oscillate. Comared to Fig., big difference can be seen in lower frequencies, where the drift force in two-layer fluids becomes negative in the incident wave of surface-wave mode over a certain range of frequency; which can be attributed to a larger value of T in the calculation formula of (5). It can be said that the results for γ =.7 are almost the same as those for γ =.9 and for a single-layer fluid of h =.d at frequencies of >.6 which is higher than the resonant frequency in heave. The wave drift force in the incident wave of internal-wave mode is always ositive but very small for larger values of γ. owever, for γ =., the nondimensional value becomes larger than double the corresonding value in the diffraction case, owing to the effect of heave motion. Figure 4 shows the results when all modes (heave, sway, and roll) of body motion are free to oscillate. In the resent comutations, the gyrational radius in roll is set to κ xx =.6b and the vertical distance between the center of gravity and the free surface is set to OG =.45b. A raid change can be seen around.5 both for the surface-wave and internal-wave modes,

12 which is obviously due to the resonance in roll. (It should be noted that the roll amlitude near the resonance is unrealistically very large because the viscous daming is not considered in the resent theory.) When the incident wave is of surface-wave mode, the wave drift forces for γ =. and.7 become negative at frequencies lower than the roll resonant frequency. owever, marked difference when comared to Fig. 3 is that the wave drift force is almost zero at very low frequencies. Another thing to be noted is that the results for γ =.9 are very close to those for a single-layer fluid over the whole frequency range including the heave and roll resonant frequencies; which imlies that a small difference in the fluid density between the uer and lower layers gives no rominent difference in the wave drift force and motion characteristics. The results shown above are for the case where a body floats in the uer fluid only and the interface is located at a relatively deeer osition. The horizontal force (like the wave drift force) may be affected by the resence of internal waves near a body, which is the case articularly when a body intersects the interface and will be studied next. 8.. Effects of the interface osition For the same Lewis-form body (b = B/ =.m and d =. m) and fixed values of h =.4m and γ =.75, only the vertical osition of the interface was changed from =.6 m to. m, including the case where the body intersects the interface. F D/.5ρ ga Surface-wave mode F D/.5ρ ga Internal-wave mode.3m.m =.6m.m γ=.75 h=.4m d=.m.m γ=.75 h=.4m d=.m =.6m.3m.m Fig. 5. on a Lewis-form body ( =.833, σ =.9) in two-layer fluids: Effect of the interface osition for the case where all body motions are fixed. F D/.5ρ ga Surface-wave mode F D/.5ρ ga Internal-wave mode.m.m =.6m.3m γ=.75 h=.4m d=.m.m γ=.75 h=.4m d=.m =.6m.3m.m Fig. 6. on a Lewis-form body ( =.833, σ =.9) in two-layer fluids: Effect of the interface osition for the case where only the heave motion is free to oscillate.

13 F D/.5ρ ga Surface-wave mode F D/.5ρ ga Internal-wave mode =.6m.3m.m.m γ=.75 h=.4m d=.m.m γ=.75 h=.4m d=.m =.6m.3m.m Fig. 7. on a Lewis-form body ( =.833, σ =.9) in two-layer fluids: Effect of the interface osition for the case where all body motions are free to oscillate. Figure 5 shows comuted results of the wave drift force for the case where all body motions are fixed, and like before the left-hand and right-hand sides are for the surface-wave and internalwave modes, resectively, of the incident wave. In the incident wave of surface-wave mode, the drift force is always ositive, and no rominent difference exists among the results for different interface ositions, excet that undulatory variation can be seen at <.5 for the case of =.3 m where the interface is located just below the bottom of the body (d =. m). On the other hand, in the incident wave of internal-wave mode, a remarkable change can be seen deending on whether a body intersects the interface. When the interface osition is deeer than the draft of a body, the wave drift force is negligibly small. owever, once a body intersects the interface, the wave drift force becomes large and increases almost linearly with resect to ; for which we may envisage that the internal incident wave will be blocked by a body and almost all waves may be reflected; that is, the coefficient of R in the calculation formula (5) is largely different deending on whether the body intersects the interface. Figure 6 shows the results when only the heave motion is free to oscillate. Obviously, owing to the heave motion, the wave drift force becomes small in frequencies lower than the heave resonant frequency, which imlies that longer incident waves transmit because the heave is free to oscillate. It should be noted, however, that the drift force at =.3 m fluctuates in lower frequencies and becomes negative in a certain range of frequency, which is due to the effect of waves with internal-wave mode, as can be conjectured from (5). On the other hand, in the incident wave of internal-wave mode, no marked difference can be seen as comared to Fig. 5, which imlies that the reflection-wave coefficient R (esecially when a body intersects the interface) is not much influenced by the heave motion and the hydrodynamic situation near the cross oint between the body and interface may be viewed locally as the diffraction roblem regardless of the heave motion. Lastly, Fig. 7 shows the results for various vertical ositions of the interface when all modes of body motion are free to oscillate. As shown in Kashiwagi et al. [], the resonant frequency in roll changes slightly deending on the osition of the interface, due mainly to the change in the roll restoring moment. Therefore, the frequency where raid variation in the wave drift force aears is slightly different deending on the vertical osition of the interface. It can be seen that the wave drift force is almost zero at lower frequencies, irresective of the interface osition. Another oint to be emhasized is that the wave drift force looks always ositive when a body intersects the interface (at =. m and.6 m), which means that the transmitted wave with internal-wave mode T in the calculation formula of (5) is relatively small. When the incident wave is of internal-wave mode, as comared to Figs. 5 and 6, slight difference can be seen around the roll resonance, but we should note that the roll amlitude near the resonance becomes unrealistically very large for lack of the viscous daming in the resent study. 3

14 9. Conclusions The wave drift force in a -D two-layer fluid of finite deth has been studied with the otentialflow assumtion. Based on the momentum and energy conservation rinciles, a comact calculation formula for the wave drift force was obtained; the key to success was to use effectively the orthogonality relations for the eigenfunctions in a two-layer fluid. Owing to the resence of the interface, for a rescribed frequency, there can exist two different incident waves with surface-wave mode (longer wavelength) and internal-wave mode (shorter wavelength), and each incident wave will be diffracted by a body into two different wave modes and hence the energy of the incident wave may be transferred from one mode to the other. The wave drift force in this rather comlicated situation was described with only one equation, which includes the coefficients of reflected and transmitted waves in a two-layer fluid. An imortant feature to be seen from this calculation formula is that a ossibility of negative drift force exists in the incident wave of surface-wave mode; which can be exerted by a large value of the transmitted wave with internal-wave mode. Numerical comutations were erformed with the boundary integral-equation method using the Green function for the two-layer fluid roblem. Comuted results of the wave drift force were shown for both incident waves with surface-wave and internal-wave modes and also for three cases where all body motions are comletely fixed, only the heave motion is free, and all body motions are free to oscillate. Furthermore, by changing the density ratio and the interface osition including the case where a body intersects the interface, those effects on the wave drift force were discussed. Main results obtained from the resent numerical study may be summarized as follows: ) When the body motions are fixed, the wave drift force seems to be ositive for all frequencies, regardless of the density ratio and the interface osition. owever, when the osition of the interface is a little deeer than the bottom of a body and the body motions are free to resond to the incident wave of surface-wave mode, the wave drift force becomes negative at frequencies lower than the resonant frequency of body motion. ) When the difference in the fluid density between the uer and lower layers is large (say γ =.), the wave drift force becomes large even in the incident wave of internal-wave mode. On the other hand, when the difference in the fluid density is small (say γ =.9), the results are very close to those for a single-layer fluid over the whole range of frequency, excet at very low frequencies when all or some of the body motions are fixed. 3) When a body intersects the interface, the body reflects most of the incident wave of internalwave mode articularly at higher frequencies, and hence the wave drift force nondimensionalized in terms of the square of wave amlitude on the interface seems to increase linearly in roortional to the square of the frequency. References. I. Ten and M. Kashiwagi, ydrodynamics of a body floating in a two-layer fluid of finite deth, Part : Radiation roblem, Journal of Marine Science and Technology 9 (3) (4) M. Kashiwagi, I. Ten and M. Yasunaga, ydrodynamics of a body floating in a two-layer fluid of finite deth, Part : Diffraction roblem and wave-induced motions, Journal of Marine Science and Technology, submitted. 3. R. W. Yeung and T. Nguyen, Radiation and diffraction of waves in a two-layer fluid. Proceedings of the nd Symosium on Naval ydrodynamics, Washington D.C. (999) Maruo, The drift of a body floating on waves, Journal of Shi Research 4 (96). 4

15 Aendix The orthogonality roerties of the eigenfunctions with resect to z in the two-layer fluid roblem are exlained in Ten and Kashiwagi [], a summary of which is given below. As exlicitly given by (), the z-deendent functions in the uer and lower layers are denoted by Z (k; z) and Z () (k; z), resectively. If necessary, the eigenfunctions corresonding to the eigenvalues k = k ( =, ) are reresented by the subscrit; e.g., Z (k ; z). Z () h (k ; z) and The orthogonality can be roven in the same way as that in the Sturm-Liouville eigenvalue roblem, and the basic equation is given as h { d Z L γ Z Z d Z h { d Z () + Z () Z () d Z () [ ] =(k k ) (k k ) γ h h Z (k ; z)z (k ; z) + w(z)z(k ; z)z(k ; z), Z () (k ; z)z () (k ; z) where { w(z) =γ, Z(k ; z) =Z (k ; z) z (A) w(z) =, Z(k ; z) =Z () (k ; z) z h Integrating by arts for the first line of (A) and substituting the boundary conditions on z =,z =, and z = h (these are the same in form as (6) (8)), one can easily rove that L = for the case of k =/ k ; that is, h (A) w(z)z(k ; z)z(k ; z) = for k =/ k. (A3) This means that there is no need to consider the integrals of cross terms between the k -wave and k -wave modes. Next, the normalization integral for the case of k = k k can be obtained by taking the limit of k k in (A). Substituting k = k + δk in (A), considering the Taylor exansion with resect to k, and retaining only the term of O(δk), the desired result for the normalization integral can be derived and exressed in the form h F k w(z) { Z(k; z) γ = K [ {Z ] + kh [ {Z ] () k z= γ z=h + εk [ {Z + γ ( + K ε k K ) {Z + k K Z Z ] = K (K chkh k shkh) + kh k γk sh kh + ε [ h ( k γ k K + ) ] (K ) (K k ) chkh k shkh + γ sh kh K K. (A4) z= 5