On freely floating bodies trapping time-harmonic waves in water covered by brash ice
aa r X i v : . [ phy s i c s . f l u - dyn ] D ec On freely floating bodies trappingtime-harmonic waves in watercovered by brash ice
Nikolay Kuznetsov and Oleg Motygin
Laboratory for Mathematical Modelling of Wave Phenomena,Institute for Problems in Mechanical Engineering, Russian Academy of Sciences,V.O., Bol’shoy pr. 61, St. Petersburg 199178, Russian FederationE-mail: [email protected] , [email protected]
Abstract
A mechanical system consisting of water covered by brash ice and a bodyfreely floating near equilibrium is considered. The water occupies a half-spaceinto which an infinitely long surface-piercing cylinder is immersed, thus allowingus to study two-dimensional modes of the coupled motion which is assumedto be of small amplitude. The corresponding linear setting for time-harmonicoscillations reduces to a spectral problem whose parameter is the frequency.A constant that characterises the brash ice divides the set of frequencies intotwo subsets and the results obtained for each of these subsets are essentiallydifferent.For frequencies belonging to a finite interval adjacent to zero, the total energyof motion is finite and the equipartition of energy holds for the whole system.For every frequency from this interval, a family of motionless bodies trappingwaves is constructed by virtue of the semi-inverse procedure. For sufficientlylarge frequencies outside of this interval, all solutions of finite energy are trivial.
This paper continues the rigorous study (initiated in [3]) of a freely floating rigidbodies trapping time-harmonic waves in an inviscid, incompressible, heavy fluid, saywater (see also [6, 7, 8] and [4]). We consider the infinitely deep water in irrotationalmotion bounded from above by a free surface unbounded in all horizontal directions,but unlike the cited papers dealing with the open surface, we assume here that itis totally covered with the brash ice. The body is supposed to be an infinitely longcylinder which allows us to consider two-dimensional modes orthogonal to cylinder’sgenerators. We also assume that the body is surface-piercing and unaffected by allexternal forces (for example due to constraints on its motion) except for gravity. Themotion of the whole system is supposed to be of small amplitude near equilibrium,and so the linear model developed by John [2] is used to describe the coupled motion1 ✻ F Fy + b + a − a − bB − ballast ballast B + S − S + xW F Figure 1: A definition sketch of the cylinder cross-section with immersed parts denotedby B − and B + ; their wetted boundaries are S − and S + respectively. The cross-sectionof the infinitely thin layer of the brash ice covering the free surface is denoted by F ;it consists of three parts two of which lie on the x -axis outside | x | > a and the thirdone is between x = − b and x = + b ; W is the cross-section of the water domain.of water and body. However, the free-surface boundary condition must be amendedto take into account the presence of the brash ice covering the water. Such a conditionwas proposed by Peters [10], who considered the brash ice as an infinitely thin matwhose particles do not interact; that is, the only forces acting on it are those due togravity and the pressure of the water from below (see also [1]).Our aim is twofold: first, we apply the so-called semi-inverse procedure for theconstruction of motionless two-dimensional bodies each trapping a time-harmonicmode; that is, the water covered by the brash ice and the body oscillate at the samefrequency, whereas the energy of water and ice motion is finite. We recall that atrapped mode in two and three dimensions describes a free oscillation of the systemthat has finite energy (see [9], p. 17). This should be distinguished from edge wavesand waves trapped around an array of cylinders.It should be mentioned that the same semi-inverse procedure was used by Kuzne-tsov [3] for obtaining two-dimensional trapping bodies that are motionless in the openwater (it is outlined in § Let the Cartesian coordinate system ( x, y ) in a plane orthogonal to the generators of afreely floating infinitely long cylinder be chosen so that the y -axis is directed upwards,whereas the mean free surface of the water intersects this plane along the x -axis, and2o the cross-section W of the water domain is a subset of R − = { x ∈ R , y < } . Let b B denote the bounded two-dimensional domain whose closure is the cross-section ofa floating cylinder in its equilibrium position (see figure 1).We suppose that b B \ R − — the part of the body located above the surface ofwater covered by the brash ice — is a nonempty domain, whereas the immersed part B = b B ∩ R − is the union of a finite number of domains (in particular, it may bea single domain). Thus, D = b B ∩ ∂ R − consists of the same number of nonemptyintervals of the x -axis; see figure 1, where D = { x ∈ ( − a, − b ) ∪ ( b, a ) , y = 0 } in thecase of two immersed parts. We suppose that W is R − \ B . Furthermore, we assumethat W is a Lipschitz domain, and so the unit normal n pointing to the exterior of W is defined almost everywhere on ∂W . Finally, we denote by S = ∂ b B ∩ R − the wettedcurve (the number of its components is equal to the number of immersed domains),whereas F = ∂ R − \ D is the infinitely thin layer of the brash ice covering the freesurface at rest.To describe the small-amplitude coupled motion of the system it is standard toapply the linear setting, in which case the first-order unknowns are used. These arethe velocity potential Φ( x, y ; t ) and the vector column q ( t ) describing the motion ofthe body. This vector has the following three components: • q and q are the displacements of the centre of mass in the horizontal and verticaldirections respectively from its rest position (cid:0) x (0) , y (0) (cid:1) ; • q is the angle of rotation about the axis that goes through the centre of massorthogonally to the ( x, y )-plane (the angle is measured from the x - to the y -axis).On the surface F , the brash ice is characterised by a non-negative function σ equalto the ratio of the local area density of the brash ice to the constant volume densityof the water. The values of σ are less than or equal to one, and we assume that σ isconstant throughout F ; the value of this constant is defined by the water salinity.We omit relations governing the general time-dependent motion (see the detailsin [3]) and turn directly to the time-harmonic oscillations of the system for whichpurpose we use the ansatz (cid:0) Φ( x , y, t ) , q ( t ) (cid:1) = Re (cid:8) e − i ωt (cid:0) ϕ ( x , y ) , i χ (cid:1)(cid:9) , (1)where ω > ϕ ∈ H loc ( W ) is a complex-valued functionbounded at infinity and χ ∈ C . Then the governing relations for (cid:0) ϕ, χ (cid:1) are asfollows: ∇ ϕ = 0 in W ; (2) ∂ y ϕ = ν ( ϕ + σ∂ y ϕ ) on F, where ν = ω /g ; (3) ∂ n ϕ = ω χ T N (cid:16) = ω X N j χ j (cid:17) on S ; (4) ∇ ϕ → y → −∞ ; (5) ω Eχ = − ω Z S ϕ N d s + g Kχ . (6)3ere ∇ = ( ∂ x , ∂ y ) is the spatial gradient and g > y -axis; N = ( N , N , N ) T (the operation T transforms a vector row into a vector column and vice versa), where ( N , N ) T = n , N = (cid:0) x − x (0) , y − y (0) (cid:1) T × n and × stands for the vector product. In the equationsof the body motion (6), the 3 × E = I M I M
00 0 I M and K = I D I Dx I Dx I Dxx + I Sy . (7)The positive elements of the mass/inertia matrix E are I M = ρ − Z b B ρ ( x, y ) d x d y and I M = ρ − Z b B ρ ( x, y ) h (cid:16) x − x (0) (cid:17) + (cid:16) y − y (0) (cid:17) i d x d y, where ρ ( x, y ) ≥ ρ > K are I D = Z D d x > , I Dx = Z D (cid:0) x − x (0) (cid:1) d x,I Dxx = Z D (cid:0) x − x (0) (cid:1) d x > , I Sy = Z S (cid:0) y − y (0) (cid:1) d x d y. It should be noted that the matrix K is symmetric.First, we suppose that νσ ∈ [0 , σ is a given constant. Under this assumption, the problem is studied in §§ νσ ≥ §
5. For νσ ∈ [0 ,
1) the boundary condition (3)is equivalent to ∂ y ϕ = ν ( σ ) ϕ on F, where ν ( σ ) = ν/ (1 − νσ ) > ν. (8)This boundary condition is of the same form as in the case of the open water whenthe coefficient ν stands on the right-hand side instead of ν ( σ ) . The parameter ν hasthe following expression in terms of ν ( σ ) : ν = ν ( σ ) / (cid:0) σν ( σ ) (cid:1) < ν ( σ ) . As in the problem describing waves on the open water, it is natural to complementthe Laplace equation (2) and the boundary condition (8) by the following radiationcondition (it means that the potential (1) describes outgoing waves): Z W ∩{| x | = b } (cid:12)(cid:12) ∂ | x | ϕ − i ν ( σ ) ϕ (cid:12)(cid:12) d s = o (1) as b → ∞ . (9)In relations (4), (6), (8) and (9), ω is a spectral parameter (into (8) and (9) it isinvolved through ν ( σ ) ), which is sought together with the eigenvector ( ϕ, χ ).4ince W is a Lipschitz domain and ϕ ∈ H loc ( W ), it is natural to understand theproblem, namely relations (2), (4) and (8), in the sense of the integral identity Z W ∇ ϕ ∇ ψ d x d y = ν ( σ ) Z F ϕ ψ d x + ω Z S ψ N T χ d s, (10)which must hold for an arbitrary smooth ψ having a compact support in W , whereasthe remaining conditions (5) and (9) specify the behaviour of ϕ at infinity.The problem formulated above must be augmented by the following subsidiaryconditions concerning the equilibrium position (see [2]): • I M = R B d x d y (Archimedes’ law — the mass of the displaced liquid is equal to thatof the body); • R B (cid:0) x − x (0) (cid:1) d x d y = 0 (the centre of buoyancy lies on the same vertical line as thecentre of mass); • the 2 × K ′ that stands in the lower right corner of K is positive definite.The last of these requirements yields the stability of the body equilibrium position,which is understood in the classical sense that any instantaneous infinitesimal dis-turbance causes the position changes which remain infinitesimal, except for purelyhorizontal drift, for all subsequent times. It is known (see, for example, [5, § ϕ ( x, y ) = A ± ( y ) e i ν ( σ ) | x | + r ± ( x, y ) , | r ± | , |∇ r ± | = O (cid:0) [ x + y ] − (cid:1) as x + y → ∞ . (11)Moreover, the following equality holds for the coefficients ν ( σ ) Z −∞ (cid:0) | A + ( y ) | + | A − ( y ) | (cid:1) d y = − Im Z S ϕ ∂ n ϕ d s. (12)Assuming that (cid:0) ϕ, χ (cid:1) is a solution of the problem formulated in section 2, werearrange the last formula by virtue of the coupling conditions (4) and (6), thusobtaining (see details in [4], § ν ( σ ) Z −∞ (cid:0) | A + ( y ) | + | A − ( y ) | (cid:1) d y = Im n ω χ T Eχ − g χ T Kχ o . (13)In the same way as in [6, 7], this yields the following assertion about the kinetic andpotential energy of the water motion. 5 roposition 1. Let (cid:0) ϕ, χ (cid:1) be a solution of problem (2) – (6) , then Z W |∇ ϕ | d x d y < ∞ and ν ( σ ) Z F | ϕ | d x < ∞ , (14) that is, ϕ ∈ H ( W ) . Moreover, the following equality holds: Z W |∇ ϕ | d x d y + ω χ T Eχ = ν ( σ ) Z F | ϕ | d x + g χ T Kχ . (15)Here the kinetic energy of the water/body system stands on the left-hand side,whereas we have the potential energy of the coupled motion on the right-hand side.The latter takes into account that the water is covered by the brash ice, and so thelast formula generalises the energy equipartition equality valid when a body is freelyfloating in the open water.Proposition 1 shows that if ( ϕ, χ ) is a solution with complex-valued components,then its real and imaginary parts separately satisfy the problem. This allows us toconsider ( ϕ, χ ) as an element of the real product space H ( W ) × R in what follows;an equivalent norm in H ( W ) is defined by the sum of two quantities (14). Definition 1.
Let the subsidiary conditions concerning the equilibrium position (see §
2) hold for the freely floating body b B . A non-trivial real solution ( ϕ, χ ) ∈ H ( W ) × R of the problem (10), (6) and (9) is called a mode trapped by this body, whereasthe corresponding value of ω is referred to as a trapping frequency . In [3], a semi-inverse procedure was used for the construction of motionless bodiesfreely floating in the open water of infinite depth and trapping a two-dimensionalmode at the frequency ω given arbitrarily. The idea of the procedure is to seekbodies for a prescribed trapped mode. Subsequently, this approach was developedin [7, 8], where various axisymmetric trapping bodies, motionless and heaving, wereconstructed as well as sets of multiple bodies some of which are motionless, whereasthe others heave. In this section, we first apply the same idea to construct a familyof motionless bodies trapping waves in the water covered by the brash ice and thenconsider the effect of the brash ice by comparing these bodies with those trappingwaves in the open water. Let ω > ν = ω /g satisfies the inequality ν < σ − , where theconstant σ > §§ ν changed to ν ( σ ) = ν/ (1 − νσ ).Let us consider the following motionless mode ( ϕ , ( , T ), where ∈ R denotesthe zero-displacement vector, 0 stands for the zero angle of rotation and ϕ ( x, y ) = g φ ( ν ( σ ) x, ν ( σ ) y ) /ω , whereas the non-dimensional velocity potential φ is as follows: φ ( ν ( σ ) x, ν ( σ ) y ) = Z ∞ k e kν ( σ ) y k − h sin k ( ν ( σ ) x − π ) − sin k ( ν ( σ ) x + π ) i d k . (16)Here, the integral is understood as usual improper integral because its integrand isbounded; indeed, the location of zeroes coincides for the denominator and numerator.Moreover, we have that φ ( ν ( σ ) x, ν ( σ ) y ) = (cid:16) ν ( σ ) (cid:17) − h G x ( x, y ; − π/ν ( σ ) , − G x ( x, y ; π/ν ( σ ) , i , where G ( x, y ; ξ, η ) is Green’s function of the time-harmonic water-wave problem (see[5], § G are described). Therefore, ∂ y φ − ν ( σ ) φ = 0 for y = 0 and ν ( σ ) x = ± π, (17)and φ ( x, y ) = O (cid:0) [ x + y ] − (cid:1) , |∇ φ ( x, y ) | = O (cid:16) [ x + y ] − / (cid:17) (18)as x + y → ∞ . These estimates yield that the kinetic and potential energy is finitefor φ in every domain away from the singularities of this function.The next crucial point of the inverse procedure is to use streamlines correspondingto the velocity potential (16) in order to define two immersed contours of a freelyfloating trapping body that is symmetric about the y -axis and has x (0) = 0. Thelast property is guaranteed by a proper choice of density distribution which is alwayspossible for a symmetric body. Moreover, it is possible to choose this distribution sothat y (0) is sufficiently close to the ordinate of the lowest points of the body, thusyielding that all of the subsidiary conditions hold.Taking a harmonic conjugate to φ in the form ψ ( ν ( σ ) x, ν ( σ ) y ) = Z ∞ k e kν ( σ ) y k − h cos k ( ν ( σ ) x − π ) − cos k ( ν ( σ ) x + π ) i d k , (19)we see that this stream function decays at infinity. Let us list several other propertiesof ψ (see their proof in [5], pp. 178–179) used for construction of a family of bodiestrapping the mode ( ϕ , ( , T ). Since ψ is an odd function of x , we formulate theseproperties only for x >
0. 7 ( σ ) xν ( σ ) y ballast S − ballast S + Figure 2: Level lines of the stream function ψ plotted in non-dimensional coordinates.An example of the wetted contours bounding a motionless trapping body is shownin bold and denoted by S − and S + ; other pairs of symmetric streamlines can also beconsidered as boundaries of immersed parts of trapping bodies. In order to contractthe width of the figure, a reduced horizontal scale is applied on the interval (-2.5, 2.5).The trace ψ ( ν ( σ ) x,
0) has only one positive zero ν ( σ ) x ∈ (cid:0) π , π (cid:1) and ψ ( ν ( σ ) x, < , ν ( σ ) x ) . Moreover, this function increases monotonically from 0 to + ∞ on (cid:2) ν ( σ ) x , π (cid:1) and de-creases monotonically from + ∞ to 0 on ( π, + ∞ ). Therefore, for every non-dimensional d > S + = (cid:8) ( ν ( σ ) x, ν ( σ ) y ) ∈ R − : x > , y < , ψ ( ν ( σ ) x, ν ( σ ) y ) = d (cid:9) (see the right part of figure 2) connects two points that lie on the positive ν ( σ ) x -axison either side of the point ( π,
0) at which ψ is infinite.Thus, for the chosen ω and every d > B + between S + = n ( x, y ) ∈ R − : ( ν ( σ ) x, ν ( σ ) y ) ∈ S + o and the x -axis serves as the right immersed part of a single motionless trapping body b B obtained by connecting B + with the symmetric domain B − as shown in figure 2for the corresponding domains in non-dimensional variables. Indeed, the Cauchy–Riemann equations imply that ϕ satisfies the homogeneous Neumann condition on S = S − ∪ S + , and so the coupling condition (4) is fulfilled for the mode ( ϕ , ( , T ).The coupling condition (6) is also fulfilled for this mode because it reduces to theequality Z S − ∪ S + ϕ n y d s = 0 . Indeed, the other two equalities are trivial by the symmetry of S − ∪ S + and the factthat the corresponding integrands are odd functions of x . The proof that the lastintegral vanishes is based on the second Green’s formula and the asymptotic formulae(18) (see details in [3]). 8 .2 The effect of the brash ice on motionless trapping bodies The procedure described in § ν ( σ ) x, ν ( σ ) y ) to ( νx, νy ) for which purpose the equality ν = ν ( σ ) / (1 + σν ( σ ) ) serves.Rescaling can be realised in two different manners — by keeping the area of the im-mersed part fixed and by keeping the length of D fixed. The corresponding results areshown in figures 3 and 4 respectively, where three values of frequency are consideredto demonstrate the effect; the frequency is characterised by the product νσ .In figures 3 and 4 (a)–(c), examples of symmetric bodies trapping waves in thewater covered by the brash ice are shown. The bodies are represented by the bound-aries of their right immersed parts; see the left contour in each figure (cf. figure 2,where an example of the whole body is shown in the non-dimensional coordinates( ν ( σ ) x, ν ( σ ) y )). The location of the body that traps waves in the open water at thesame frequency is indicated for comparison; it is plotted in bold (see the contour onthe right-hand side of each figure). Both bodies have either the same area of theimmersed part (the scaling adopted in figure 3) or the same length of D (the scalingadopted in figure 4). The dashed lines in figures 3 (b) and 4 (b) show the contoursfrom which the left ones are obtained by the scaling ( ν ( σ ) x, ν ( σ ) y ) ( νx, νy ).Thus, figures 3 (a)–(c) demonstrate the following properties of the obtained sym-metric trapping bodies with the fixed immersed area. The spacing between two im-mersed parts is less for bodies that trap waves in the water covered by the brash icethan for bodies that trap waves in the open water. Moreover, this spacing decreasesas the frequency of trapped waves increases, whereas the length of D , on the opposite,increases together with the frequency.The properties of symmetric trapping bodies with the fixed length of D are asfollows; see figures 4 (a)–(c). The spacing between two immersed parts demonstratesthe same behaviour as in the other case, whereas the immersed area decreases as thefrequency increases. νσ ≥ Let b B be a freely floating body, that is, it satisfies all subsidiary conditions of §
2. Forthe matrices E and K corresponding to b B (see (7) and the following formulae thatdefine the matrices’ elements), we denote by λ ∗ the largest λ such that det( λ E − K ) =0. Proposition 2.
Let ( ϕ, χ ) ∈ H ( W ) × R be a solution of problem (2) – (6) . If ω is such that ν ≥ max { λ ∗ , σ − } , where λ ∗ corresponds to b B through E and K , then ( ϕ, χ ) is a trivial solution.Proof. Let R α,β = { ( x, y ) ∈ R − : | x | < α, y > − β } , where α, β > B ⊂ R α − ,β − . Applying the first Green’s formula, we9 .5 1.5 2.5 3.5 4.50.0 − − − − − − νxνy νxνy νxνy ( a )( b )( c )Figure 3: Examples of bodies trapping waves in the water covered by the brash icefor νσ = 1 / νσ = 1 / νσ = 3 / D and traps waves in the open waterat the same frequency. The dashed line in figure (b) shows the contour from whichthe left one is obtained by the scaling ( ν ( σ ) x, ν ( σ ) y ) ( νx, νy ).10 .5 1.5 (cid:1) (cid:0) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) νxνy νxνy νxνy ( a )( b )( c )Figure 4: Examples of bodies trapping waves in the water covered by the brash icefor νσ = 1 / νσ = 1 / νσ = 3 / ν ( σ ) x, ν ( σ ) y ) ( νx, νy ).11rite Z W ∩ R α,β |∇ ϕ | d x d y = Z S ϕ ∂ n ϕ d s + Z α − α [ ϕ ( x, y ) ∂ y ϕ ( x, y )] y =0 ,x ∈ Fy = − β d x + Z − β [ ϕ ( x, y ) ∂ x ϕ ( x, y )] x = αx = − α d y, (20)where the Laplace equation is taken into account. The assumptions imposed on ϕ yield that: 1) there exists a sequence { α k } ∞ k =1 such that α k → ∞ as k → ∞ and thelast integral with β = ∞ and α = α k tends to zero as k → ∞ ; 2) for all α > R α − α [ ϕ ( x, − β ) ∂ y ϕ ( x, − β )] d x → β → ∞ . Passing to the limit in (20),first as β → ∞ and then as α k → ∞ , we obtain Z W |∇ ϕ | d x d y − Z F ϕ ( x, ∂ y ϕ ( x,
0) d x = Z S ϕ ∂ n ϕ d s. (21)Here the integral over F vanishes when ν = σ − (in this case, the boundary condition(3) reduces to ϕ = 0 on F ); if ν > σ − , this integral is equal to ν ( σ ) R F [ ϕ ( x, d x ,which is a consequence of (8). In both cases, the left-hand side of (21) is positive fora non-trivial ϕ (in the second case, because ν ( σ ) < Z S ϕ ∂ n ϕ d s = ω χ T Z S ϕ N d s = − g − χ T [ ν E − K ] χ , and so the right-hand side term of (21) is non-positive when ν ≥ λ ∗ . Thus, thiscontradicts to the positivity of the left-hand side with a non-trivial ϕ when ν ≥ max { λ ∗ , σ − } . This completes the proof of the proposition. This note extends our previous work on motionless trapping bodies. Here, it has beenshown that there exist two-dimensional bodies with a vertical axis of symmetry andtwo immersed parts and the following properties. These bodies are freely floatingbut motionless and trap some time-harmonic modes provided their frequencies arebounded by a constant characterising the brash ice. Thus, there is a coupled motionof the water covered by this ice that does not radiate waves to infinity in the presenceof such a body. Therefore, in the absence of viscosity, this oscillation will persistforever.On the other hand, for every freely floating body there exists a bound for frequen-cies (it depends also on the constant that characterises the brash ice) such that nomodes are trapped at the frequencies exceeding this bound. Unlike uniqueness the-orems known hitherto and concerning time-harmonic waves in the presence of freelyfloating bodies, this result does not involve any geometric restrictions.12 eferences [1]
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