Cooker, M.J.

Pǎrǎu, E.

Vanden-Broeck, J.-M.

English

University of East Anglia digital repository

Three-dimensional waves in a two-fluid system generated by amoving pressureEmilian I. Pa˘ra˘u, Jean-Marc Vanden-Broeck, Mark J. CookerSchool of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UKe-mail: e.parau@uea.ac.uk1. IntroductionWe consider a three-dimensional steady wave problem in which a distribution of pressuremoves at constant velocity U on the interface between two fluids, with a lighter fluid lying abovea heavier one. The lower fluid is of infinite depth and the upper fluid layer is bounded above bya free surface. This models, in an inverse way, interfacial flows past submerged bodies on theinterface. The pressure can also be applied on the free surface, instead of the interface. Thethree dimensional problem is formulated as a nonlinear integro-differential equation by usingthe Green’s identity in each layer and the dynamic boundary conditions.In the last half century there were a number of research papers considering forced waveswhich propagate at the interface of two-fluids (see for example Hudimac [4], Crapper [2], Kellerand Munk [5], Yih and Zhu [10], Tulin and Miloh [7], Avital and Miloh [1], Wei et al [8], Yeungand Nguyen [9]). Most of them are studying the linear problem. It was shown that the wavesystem at the interface or at the free surface is determined by a surface-wave mode and aninternal-wave mode.2. Formulation and numerical schemeThe fluids are assumed to be inviscid and incompressible and the flow to be irrotational inboth layers. There is no shear between the layers and the free-surface and interfacial tensionsare neglected. The subscript 1 refers to the lower fluid, and the subscript 2 refers to the upperlayer. The lower layer is of infinite depth and the upper fluid, which is bounded above by afree surface, has a rest equilibrium thickness h. We choose a frame of reference moving withthe disturbance. We introduce Cartesian coordinates x, y, z with the z-axis directed verticallyupwards with z = 0 in the still interface and the x-axis in the opposite direction of the velocityU and we denote by z = ζ(x, y) the position of the interface and by z = ζs(x, y)+h the equationof the free-surface.The governing equations in each layer are∆Φi = 0 , i = 1, 2 , (1)where Φi is the velocity potential in layer i. On the interface z = ζ(x, y) the full nonlinearkinematic and dynamic conditions are appliedΦixζx + Φiyζy = Φiz, i = 1, 2, (2)12ρ1(Φ21x + Φ21y + Φ21z)−12ρ2(Φ22x + Φ22y + Φ22z) + (ρ1 − ρ2)gζ + p =12(ρ1 − ρ2)U2. (3)Here ρi are the densities, g is the acceleration due to gravity, and p is the pressure distributionwhich models the disturbance at the interface.On the free-surface z = ζs(x, y) + h we have again the kinematic and dynamic conditionsΦ2xζx + Φ2yζy = Φ2z, (4)12(Φ22x + Φ22y + Φ22z) + g(ζs + h) =12U2 + gh. (5)The velocity and radiation boundary conditions at infinity are(Φ1x, Φ1y, Φ1z) → (U, 0, 0) for z → −∞ (6)no waves for x → −∞. (7)We present results by choosing the pressure asp(x, y) = P0eL2x2−L2+L2y2−L2 , |x| < L and |y| < L0 otherwise.Here P0 is a constant and L defines the size of the support of the pressure. We introducedimensionless variables by using U as the unit velocity and L as the unit length.Combining equations (2) and (3), respectively (4) and (5), and using the chain rule weobtain12(1 + ζ2x)φ21y + (1 + ζ2y)φ21x − 2ζxζyφ1xφ1y1 + ζ2x + ζ2y−R2(1 + ζ2x)φ22y + (1 + ζ2y)φ22x − 2ζxζyφ2xφ2y1 + ζ2x + ζ2y+1− RF 2ζ + εP =1− R2, (8)12(1 + ζ2sx)φ2sy + (1 + ζ2sy)φ2sx − 2ζsxζsyφsxφsy1 + ζ2sx + ζ2sy+ζsF 2=12(9)where φi(x, y) = Φi(x, y, ζ(x, y)), φs = Φ2(x, y, ζs(x, y) + H), F = U/(gL)1/2 is the Froudenumber, R = ρ2/ρ1 is the density ratio, H = h/L is the relative thickness of the upper layerand ε =P0ρ1U2is the dimensionless magnitude of the pressure.The numerical scheme is an extension to a two-fluid system of that used by Pa˘ra˘u andVanden-Broeck [6] for the computation of forced gravity waves in deep water. It is based ona boundary integral equation method introduced by Forbes [3] for three-dimensional gravityfree-surface flows past a source.The formulation involves applying in each fluid Green’s second identity for the functionsΦi − x (i = 1, 2) and G, where G(P, P∗) is the three dimensional free space Green functionG(P, P ∗) =14pi1([x− x∗]2 + [y − y∗]2 + [z − z∗]2)1/2, (10)for the volumes V1 and V2 and P = (x, y, z), P∗ = (x∗, y∗, z∗) ∈ SI . The volume V1 consistsof a large-radius hemisphere bounded above by the interface SI , except for a small hemispherearound the point P ∗ and the volume V2 consists of a large-radius cylinder bounded by thefree surface SF and the interface SI , except a small hemisphere around the point P∗. Anotherintegral equation is obtained when P ∗ is on the free-surface SF and we apply the Green’ssecond identity on the volume V ′2 , which is similar with V2, the difference being that the smallhemisphere excepted around the point is now on the free surface. We obtain the followingequations12(Φ1(P∗)− x∗) =∫SI(Φ1(P )− x)∂G(P, P ∗)∂n1−G(P, P ∗)∂(Φ1(P )− x)∂n1dS for P ∗ ∈ SI ,12(Φ2(P∗)− x∗) =∫SI(Φ2(P )− x)∂G(P, P ∗)∂n2−G(P, P ∗)∂(Φ2(P )− x)∂n2dSP+∫SF(Φ2(P )− x)∂G(P, P ∗)∂ns−G(P, P ∗)∂(Φ2(P )− x)∂nsdSP , for P∗ ∈ SI and P∗ ∈ SF ,where ni (i = 1, 2) is the normal vector at the interface pointing into fluid i and ns is thenormal vector at the free surface.The integro-differential equations are projected onto the Oxy plane and the singularities onthe integrals may be isolated by addition and subtraction of a quantity, which can be evaluatedin closed form (see Forbes [3] for details).For the numerical scheme we truncate the intervals −∞ < x < ∞ and 0 < y < ∞ tox1 < x < xN , and y1 < y < yM and introduce the mesh points xk = (k − 1)∆x, k = 1, . . . , Nand yj = (j − 1)∆y, j = 1, . . . , M . The 5NM unknowns are the values of ζx, ζsx, φ1x, φ2x, φsxat the mesh points. The integrals and the Bernoulli equation are evaluated at the points(xk+1/2, yj), k = 1, ..., N − 1, j = 1, ..., M so we have 5(N − 1)M equations. Another 5Mequations are obtained from the radiation condition. The values of ζ and ζs are obtained byintegrating ζx and ζsx with respect to x by the trapezoidal rule. The values of the dependentvariables at midpoints are found by interpolation and the derivatives are calculated by usingfinite differences. The 5NM nonlinear equations are solved by a modified Newton’s method.In most computations, we used N = 50, M = 20, ∆x = ∆y = 0.4.3. ResultsThe numerical scheme calculated solutions for several values of the Froude number F ,density ratio R, the relative thickness of the upper layer H, and of the pressure parameterε. The pressure was applied either at the interface, or on the free surface. The wave patternsfor F = 0.7, R = 0.9, H = 0.3, ε = 1 in the two cases are shown in Figure 1. As noted byYeung and Nguyen [9] on their linear study, the surface wave elevation and the interfacial waveelevation can each be composed of different wave systems: divergent waves and transverse wavesdue to the surface wave mode and the divergent and transverse waves due to the internal wavemode.−5 05 1015 20−10−50510−0.2−0.100.10.20.30.4xyz−5 05 1015 20−10−50510−0.3−0.2−0.100.10.20.30.4xyzFigure 1: (a) The free surface waves and the interfacial waves generated by the applied pressuremoving on the free surface inside |x| < 1 and |y| < 1. Here F = 0.7, R = 0.9, H = 0.3, ε = 1.(b) As (a) but with the waves generated by the applied pressure moving on the interface.When H is increased, the amplitude of the waves on the interface (when the pressureis applied on the free surface), or on the free surface (when the pressure is applied on theinterface) decreases.The influence of the surface-wave mode on the interfacial waves can be observed when thefree surface is replaced by a rigid lid. In Figure 2 we compared the contours of the interfacialwaves for the two cases.−2 0 2 4 6 8 10 12 14 16−8−6−4−202468xyFigure 2: The interfacial waves generated by a pressure moving on the interface, when the freesurface is present (lower half) and when the rigid-lid approximation is used (upper half). Thecontours are curves of constant vertical displacement for the same heights in both halves.AcknowledgmentsThis work was supported by EPSRC, Grant GR/S47786/01.References[1] E. Avital and T. Miloh, On an inverse problem of ship-induced internal waves, OceanEngineering 26 (1998), 99–110.[2] G.D. Crapper, Ship waves in a stratified ocean, J. Fluid Mech. 29 (1967), 667–672[3] L.K. Forbes, An algorithm for 3-dimensional free-surface problems in hydrodynamics, Jour-nal of Computational Physics 82 (1989), 330–347.[4] A.A. Hudimac, Ship waves in a stratified ocean, J. Fluid Mech. 10 (1961), 229–243.[5] J.B. Keller and W.H. Munk, Internal wave wakes of a body moving in a stratified fluids,Phys. Fluids 13 (1970), 1425–1431.[6] E.I. Pa˘ra˘u & J.-M. Vanden-Broeck, Nonlinear two and three dimensional free surface flowsdue to moving disturbances, Eur. J. Mech. - B/Fluids 21 (2002), 643–656.[7] M. Tulin & T. Miloh, Ship internal waves in a shallow thermocline: the supersonic case,Proc. 18th Symp. Naval Hydrodynamics, National Academy Press (1991), 567–581.[8] G. Wey, D. Lu and S. Dai, Waves induced by a submerged moving dipole in a two-layerfluid of finite depth, Acta Mecanica Sinica 21 (2005), 24–31.[9] R.W. Yeung and T.C. Nguyen, Waves generated by a moving source in a two-layer oceanof finite depth, J. Engng. Mathematics 35 (1999), 85–107.[10] C.S. Yih and S. Zhu, Patterns of ship waves, Quart. Appl. Math. 47 (1989), 17–33.

Three-dimensional waves in a two-fluid system generated by a moving pressure

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