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Three-Dimensional Instability Modes of the Wake Far Behind a Ship G. Triantafyllou (Massachusetts Institute of Technology, USA) 1 Abstract The three-dimensional linear stability of the viscous wake far behind a ship is investigated. The Euler equations are linearized around the time-average flow in the wake, and the stability problem is formulated as an eigenvalue problem for waves travelling parallel to the course of the ship. It is shown that the complex phase velocity of the unstable waves satisfies Howard's semi-circle theorem. For a self-similar velocity profile, a numerical solution is obtained by expanding the perturbation pressure in a Fourier series and solving a set of simultaneous ordinary differential equations. It is found that the wake becomes unstable in its antisymmetric pressure mode for a range of wavenumbers less than a "cut-off" value. In the zero Froude number limit, the frequency and phase- velocity of the unstable gravity waves are determined entirely by the characteristics of the shear flow in the wake, a fact allowing the derivation of a simple approximation for the eigenvalues and eigenvectors. As the Froude number is increased, the growth-rates of the unstable waves, and the "cut-off" value are reduced, indicating a stabilizing effect on this mode. In the infinite Froude number limit the wake becomes unstable in a different mode. The wake does not exhibit self-excited behaviour, because the instability of the shear flow is of the convective type at all Froude numbers. External noise, like ambient waves in the ocean, can drive the wake instability producing spatially growing waves. The free-surface manifestation of the spatially unstable waves exhibits a characteristic staggered pattern of alternating "hills" and "valleys". 2 Introduction The best known feature of the flow behind a ship is the Kelvin wave pattern. The Kelvin wave pattern has been extensively studied, because it is a very significant source of ship resistance at high speeds. The viscous wake of the ship has received much less attention, since it is assumed that it is for most ships thin, and its influence on the wavemaking of the ship has routinely been neglected, with some notable exceptions (Tatinclaux, 1970, Peregrine, 1971). In recent years, however, the viscous wake of the ship has attracted a considerable amount of attention, both as a basic fluid mechanical problem, and in connection with the problem of wake imaging. As aerial pictures of the ocean have revealed, viscous wakes of ships are visible at very large distances behind the ship (see, among others, Peltzer et al., 1978, Milgram, 1988, and Skop et al., 1990). Thus, the viscous wake, even though relatively thin, leaves a very persistent "trace" on the ocean surface, and offers an effective means of ship detection. The problem has several different aspects, including the backscatter of electromagnetic waves from the ocean surface (Valenzuela, 1978), and is currently extensively studied. From the fluid mechanics point of view, which is mainly of interest in the present paper, investigations of the interaction of vertical flows with a free surface have revealed several interesting new properties: Benney and Chow, 1986, Sarpkaya, 1986, Lugt, 1987, Oikawa et al., 1987, Tryggvason, 1988, Willmarth et al. 1989, Bernal and Kwon, 1989, Triantafyllou and Dimas, 1989 Liepmann, 1990. The basic hydrodynamics of wake/free surface interactions are not understood well enough yet to provide a full description of the complex phenomena involved, nor an explanation of the aforementioned persistence of ship wakes. A solution of the problem from first principles, through direct simulation of the Navier-Stokes equations, is still impossible, owing to the combined complexity provided by the very high value of the Reynolds numbers of ships, typically 109, and the presence of a moving boundary, the free surface, which renders the computational domain time-dependent. It appears therefore that the problem has to be approached in successive stages. In this paper a specific aspect of the wake/free surface interaction is addressed, namely the linear hydrodynamic stability of wake, seen as a three-dimensional shear flow. It has been well known that in supercritical transitions, linear theory can determine whether a certain flow state is unstable or not. In recent years, it has become increasingly clear that linear theory can also provide a good description of the "shape" of the unsteady flow patterns that result from the instability, whereas the amplitude of the patterns is determined by non-linear effects. (See for instance Koch, 1985, Triantafyllou et al., 1986, Triantafyllou et al., 1987, Chomaz et al., 1988, Unal and Rockwell, 1988, Karniadakis and Triantafyllou, 1989, Hanneman and Oertel, 1989, Triantafyllou and Karniadakis, 1990). Linear theory has thus become a very useful conceptual tool in interpreting the physics of unsteady viscous flows. A fundamental concept in the linear instability theory in media that are unbounded in the direction of propagation of the instability waves is the distinction between absolute and convective instabilities (see the review article by Bers, 1983). Absolutely unstable flows are self-excited, and a localized perturbation leads to growing motions at any fixed location in space. Convectively unstable flows on the other hand remain steady in a noise-free environment, because all localized perturbations are convected away. It is interesting to investigate to what extent these concepts can elucidate the problem of shear flow/free surface interaction. For the two-dimensional wake/free surface interaction problem, it has been recently shown (Triantafyllou & Dimas, 1989) that the vicinity of a free surface drastically alters the instability properties of two-dimensional shear flows, and renders an absolutely unstable flow convectively unstable. As a result, high Reynolds number wakes of floating objects remain steady at low Froude numbers, and have the fonn of steady recirculating flows. Few things have been known for the considerably more complex three-dimensional problem, studied here. We consider the space- time evolution of perturbations around the mean flow in the wake. The perturbations have the form of waves that propagate parallel to the course of the ship, and have an eigenfunction type of dependence in the other two directions. The presence of the free 553
surface is acknowledged through the kinematic and the dynamic boundary conditions. An eigenvalue problem is thus obtained for the frequency which depends parametrically on the Froude number of the flow. The eigenvalues and eigenvectors are solved for numerically. From the computed eigenvectors, the shape of the free-surface manifestations of the instability waves is determined. The issues that are of interest here are: (i) The shape of the unsteady patterns that result from the wake/free surface interaction, and (ii) Whether these patterns can be self-excited (which, as mentioned before, is related to the absolute versus convective instability distinctions). 3 Three-dimensional shear flow/free surface interactions In this section we discuss the formulation of the shear flow/free surface interaction problem in general. Let x,y,z be a system of coordinates, with the x-axis parallel to the direction of the flow, the z-axis parallel and opposite to the direction of gravity, and the y-axis perpendicular to the other two. The unit vectors along the x, y, z axes are i, j, k respectively (figure 1). For the problem of interest here, the frame of reference is fixed with respect to the ship. We assume that all velocities have been non-dimensionalized with respect to some reference velocity UOO, (in this case the speed of the ship), the pressure with respect to pu`~,2 (p is the density), and all lengths with respect to the width b of the wake. Consistent with this non-dimensionalization, the acceleration of gravity g is replaced by 1/F2, where F is the Froude number of the flow, defined as F = UOO/~. The non-dimensional Euler equations, linearized around a parallel flow U(y,z), can be written as follows (Drazin and Howard, 1966): aa +uaa )U+vvu+aap = 0 a'+Uaax )V+VP = 0 (1) where u is the component of the perturbation velocity parallel to the x-axis, v = ( v, w) is the projection of the perturbation velocity vector in the y,z plane, and p is the perturbation pressure field; also, V=(a/3y, 3/3z). In equation (1) p is the dynamic pressure, i.e. the totalpressurep minus the hydrostatic: p=p+F-'z. The incompressibility condition requires that the perturbation velocity has to be divergence-free: aU+vv = 0 ax (2) At the free surface we have the kinematic and dynamic conditions for the free-surface elevation q(x,y,t). They can be written as follows: al1 +u3~ ) = vk = w (3) a' ax p F-2 ~ (4) U<- k ~ j x 1 ~ I 1 _ _ ~ ~U(y,z) Figure 1: Definition sketch. From the momentum equation in the z-direction (the second of (1)), we have that: aW+ uaW ) = _a at ax az We use (5) to eliminate w from (3), in order to combine the two boundary conditions into a single condition for the pressure: F2(a3+Uaa )2p+3aP=0 (6) The dispersion relation of the flow can now be obtained by considering wavy perturbations, i.e. by setting u,v,w,p into the momentum and continuity equations proportional to exp(i(kx-c3t)), where ~ is the frequency and k the wavenumber. Then the momentum equations (1) become: i(kU-ce)u+vVU+iLp = 0 i(k U-~) v + Vp = 0 (7) The incompressibility condition for the perturbation velocity v = (u, v,w) is written as: iku+ V v=0 The boundary condition (6) at the free surface is written as: (8) F2(kU_ ce)2p__= 0 (9) az We also impose the condition that the perturbation decays far outside the wake, i.e. u,v,w,p - 0 when ~y2+z2 - 0O. We have used for notational simplicity in (7), (8), (9) the same symbols for the perturbation quantities in the frequency and in the time-domain; this does not cause confusion since we will work mainly in the frequency domain. We multiply the first of (7) by ik, operate on the second of (7) with V, add the two, and use the incompressibility condition (8) to obtain a single second-order partial differential equation for the dynamic pressure p: 554
(kU-~)( V2p-k2p ) -2kVpVU = 0 (10) Equation (10) subject to the boundary condition (9) plus the condition p ~ O for by 1,1 z I ~ on defines an eigenvalue problem for co, which depends parametrically on the Froude number. If for some range of real wavenumbers complex frequencies with positive imaginary parts exist, the flow is unstable. In fact, since the flow is inviscid, complex eigenvalues appear in conjugate pairs, corresponding to one growing and one decaying mode. Finally, when the flow is uniform, i.e. dU/dr=O, the basic equation (10) reduces to Laplace's equation, and from the free surface boundary condition the classical dispersion relation for deep water waves is recovered. We will now show that Howard's semi-circle theorem is valid for the complex eigenvalues. To this purpose, we first recast the basic equation (10) into the following form: V( P 2) - k2 P 2 = 0 (11) where c=~/k is the phase velocity of the wave. We multiply (11) by the complex conjugate of p, px and integrate along the y,z plane. This gives: \\ V l l 2 where Q stands for: IAdYdz(Pxv(< P 2) - k2 P ) = 0 (12) Since unstable waves have complex phase velocity, U- c does not vanish anywhere in the flow field, and the integral in (12) converges. For the first term of the integrand we write: px V ( Vp 2 ) = V ( ~ P ~ 2) - ~ U- c ~ 2 (13) We substitute into (12), and apply the divergence theorem for the integral of the first term. dydzV( P Vp )= dl n P Vp (14) IA (U C)2 J-s (U C)2 where S represents the boundary of the domain A, consisting of the free-surface and a line at infinity (figure 2), and n is the outward pointing unit vector. We use the boundary condition at the free surface, and that p tends to zero at infinity to obtain: I dYdZ(U-C)2( IVPI2+k2IPI2)I U C l-4 = A I CO = k2F2 dYIP(Y 0)12 (15) With c=cr+ici, I ci l > 0, we can separate the real and imaginary parts of (15) as follows: |AdYdz((u-cr)2-ci2)Q= (16) = k2F2J dylp(y 0)12 _co \ y ~- / S Figure 2: Integration contour in the y, z plane. | dydz(U-cr)Q = 0 (17) A Q=( IVPI2+k2lPI2)l U C 1-4 2 0 (18) Also, following Howard, 1961, we note that, if Umin,U,,~ denote, respectively, the minimum and maximum velocity in the flow field, we have: | dy dz ( U-Umin ) ( U- Um~C) Q < 0 (19) We use (16), (17) to eliminate the integrals |U2Q |UQ from the left side of (19). This yields: 0 2 | dydz(U-Umin)(U-Um~v`)Q = | dydz x A A ((Cr + Ci ) - (Umin + Um`Dc) Cr + UminUm~c ) Q + I oo + k2F2 dy IP(Y 0) 12 2 _oo 2 | dydz((Cr_ min m~X )2+c 2 A U i -U (20) which implies, since Q is positive, that: Umin+ Umax )2+C 2 ( min m )2 ~ O (21) Equation (21) is Howard's semi-circle theorem (Howard, 1961, Drazin and Howard, 1966), for the three-dimensional shear-flow free surface interaction; it states that the vector cr,ci lies within a 555
circle with center at the average of the maximum and minimum of the velocity, and radius half their difference. Equation (21) generalizes an earlier result by Yih, 1972 for the instability of two- dimensional shear flows with a free-surface. Before proceeding with the instability of the wake, it is useful to discuss briefly the zero and infinite Froude number limits of the formulation. In the former the free surface condition is reduced to ap/az = 0, whereas in the latter to p = 0. It is reasonable to assume that on the free surface Du/az=0. In this case, we can define the "double-flow" in the whole space through the extension U(y,z)=U(y,-z). Then the zero Froude number limit corresponds to an instability mode with pressure that is symmetric around the z=0 plane in the unbounded fluid, and the infinite Froude number limit to a mode with pressure that is antisymmetric around z=0. In two-dimensional flows, for instance, the zero Froude number limit corresponds to the "varicose" mode in unbounded fluid, and the infinite Froude number limit to the "sinuous" mode. For wake flows, the first is connectively unstable, and the second absolutely unstable. The instability properties of the wake as a function of the Froude number give a smooth transition from the one limit to the other (Triantafyllou and Dimas, 1989). 4 Instability of the far wake Far behind the ship, where the effects of the details of the ship hull form have diffused, we assume that the average flow has become self-similar, independent of the angle ~ = atan(z/y) in the y,z plane. We consider, in other words, a time-average velocity of the form U = U(r). The reasoning behind this assumption is that, at low Froude numbers, the time-average flow can be approximated by half of that behind the "double-body", ( a fictitious object that is twice the submerged part of the ship), which far behind the object asymptotically tends to acquire an axisymmetric form. This assumption is supported by the numerical computations of Swean, 1987, who computed the steady flow past a ship using the parabolized Navier-Stokes equations, and a K-£ model of turbulence. Swean's results suggest that indeed the steady flow tends to become self-similar; his computational results show good agreement with the experimental results of Mitra et al., 1985, and Mitra et al., 1986. Because of the self-similarity assumption, it seems natural to work in polar coordinates. Equation (10) for the perturbation p(r,0) is written in polar coordinates as follows: (kU-~)(aa P2+laaPr+ 120eP2-k2p)-2kddrU3aPr =0 (22) defined for r 2 0, -A < ~ ~ 0. For the boundary conditions on the free surface, we note that on ~ = 0, -it, we have: COs(~) a a ., Equations (22), (24), (25) define for any given k an eigenvalue problem for the frequency of; that is to say, they constitute the dispersion relation for gravity waves propagating above the wake of the ship. 5 Fourier Series Solution Because of the linearity of the problem, and the symmetry of the average flow U(r) around the plane y =0, an arbitrary perturbation can be decomposed into two parts: One in which the pressure is anti-symmetric around y = 0, referred to as Mode I, and one in which the pressure is symmetric around y = 0, referred to as Mode II. Thus, given that the free surface elevation is proportional to the value of the dynamic perturbation pressure there, Mode I disturbs the free surface in an antisymmetric manner around y = 0, whereas Mode II disturbs the free surface in a symmetric manner (figure 3). The fact that the two modes are separable facilitates the numerical solution of the problem. L / / / \ Y ' '/ / MODE I - y \J ' ~ \ / MODE 11 (23) Figure 3: Free surface elevation for Mode I (antisymmetric), and Mode II (symmetric). Consequently, the boundary condition on the free surface becomes: cos(~)(kU-cl))2F2p--aaP~ = 0 on ~ = 0, -'c (24) 5.1 Mode I Finally, we impose the condition that far from the wake the perturbation vanishes, i.e. p(r,f3) ~ O r ~ so (25) 556 We start with mode I, which satisfies the following symmetry relations: p(r,-~) = - p(r,O) aaPe(r.-~) = aP~(r,O) (26)
Since the initial flow is independent of the angle 8, it is convenient to expand p(r,8) in a cosine-series in the interval -a < ~ c 0. In accordance with the antisymmetric character of Mode I, the Fourier series will contain odd-order coefficients only. Direct substitution of a Fourier series in the basic partial differential equation (22) is not applicable, because, for non-zero values of the Froude number, p satisfies a mixed boundary condition on the free surface, and its Fourier series can not be differentiated twice. We thus make the following substitution: p(r,8) = q(r,8) + aP~(r,°) (~+2 ) (27) or, after using the free surface boundary condition (24): where in (33) Lo stands for the following operator: Lo = d + ~ d _ k2 (34) In the derivation of (33) the following result has been used: J: do ( ~ + 2 ) cos (n 8) = ~ 2 ~ (35) We also need to express G in terms of the Fourier coefficients qn. This can be done by substituting (31) into (28) and the result into (29); we obtain: pf r, 8 ) = q fir, id ) + G ( r) ( f3 +72t ~(28) G = F 2 rick U- 03~ 2 ( I, q,'+ J2C G ~(36) where G(r) is defined by: G(r) = F2r(k U- ce ) 2 p(r,O) (29) e=0,-~: The new variable q(r,8) satisfies the following conditions at a~(r,O) = aa~q(r,-~) = 0 (30) The function 0+~/2 is antisymmetric around the y=0 plane. Consequently, the new variable q can be expanded in a cosine Fourier series containing odd terms only. The Fourier series is twice differentiable with respect to ~ because of the boundary conditions (30). We write for q: 00 q(r,{~) = ~' qn( r ) cos (n 0) (31) no= Where in (31) it is implied that the summation is carried over all odd n only. The same convention applies for the rest of the section too. The coefficients qn(r) of the Fourier series in (31) are given by: qn( r) = tic ~ q(r,0) cos (nod) do (32) In order to obtain the differential equations satisfied by the coefficients qn, we substitute (28) into (10) and use the finite Fourier transform method, i.e. we multiply both sides of (10) by cos(n8), and integrate with respect to ~ from -a to 0. In an unbounded fluid this procedure yields an infinite set of uncoupled ordinary differential equations for qn(r), compeletely equivalent to those obtained for the perturbation velocity by Batchelor and Gill, 1962. In the problem considered here, however, owing to the presence of the free surface, the equations for the coefficients of the Fourier series are coupled, as follows: ( kU-o) ( Loqn - 2 qn + 2LoG ) - (33) - 2k d ( d + 2 dr ) = 0 or, after solving with respect to G: G = 2 2 qn (37) 1 - (~t/2)F rake-I) n Equation (37) expresses G(r) in terms of qn(r), but is not very appropriate for the numerical solution of the problem, since it depends non-linearly on the frequency m. In order to obtain relations between the unknown variables that are linear in in, we introduce an auxiliary unknown function H(r) defined as follows: H = F(kU-~)p(r,O) (38) Now equation (36) can be replaced by the following pair of coupled equations: H = F(kU-m)( I, qn+2G ) n G = F(kU-~)rH (40) Also, because of the antisymmetry of the free surface elevation around the plane y=O, we have the following condition for the coefficients of the Fourier series: qn(r=O) = G(r=O) = 0, n= 1,3,... (41) Equations (39), (40), and (33), subject to the conditions (41), and that the unknown functions H(r), G(r), qn(r) n = 1, 3, 5,... vanish as r ~ on, define for a given k an eigenvalue problem that depends linearly on the eigenvalue m. Consequently, if we truncate the domain to O < r < R. and use finite-differences to approximate the derivatives in (33) at specified points ri, i = 1,2,..-N, the discretized versions of equations (39), (40) and (33) define a generalized algebraic eigenvalue problem for m. The latter can be solved using a standard Q-Z algorithm. 557
5~2 Mode II Mode II satisfies the following symmetry relations: p(r,O) = p(r,-rc) (4V aaPe(r °)=-aP~(r,-~) The eigenvalue problem for Mode II can thus be formulated in a similar manner as for Mode I, with the important difference that the Fourier series has to be chosen in a way that is compatible with the symmetry conditions (42). We thus use the following substitution (instead of (28)): p(r,f3)=q(r,8)+~; aP~(r~o)(o+2)2 (43) As with Mode I, the derivative of q(r,8) satisfies the boundary conditions (30), and can be expanded in a cosine Fourier series: q(r,[~) = 2qo(r)+~ qn(r)C°S(n0) (44) n where the coefficients qn are given by (32), and the summation in (43) is carried now over all even n only. We substitute (43) into (10), and apply the finite Fourier transform technique, noting that for n even: 1 ( ~ +72` ) 2cos (n 8) do = 2 (45) Using the same procedure as for Mode I we obtain the following equations: H=F(kU-(~)(2q°+2-qn+4G) (46) G = F(kU-~)rH (47) for n = 2,4,... we have: (kU-(~) ( LoQn- 2qn+ 2LoG )- (48) -2k d ( d" + 2 d ) = 0 whereas for n=0 we have: (kU-(~) ( LoQo+6LoG + 7` 2)~2k dr ( dr + ~6 dG ) = 0 (49) in equations (47), (48), and (49) G(r)=F2r(kU-~)2p(r,0), and Lo is defined as before by equation (34). Finally, instead of (41), we have that the slope of the unknowns vanishes at r=O: qn(r=0) = ddG(r=0) = 0 n=0,2,4,... (50) and that the unknown functions H(r), G(r), and qn(r), n= 0, 2, ... vanish as r ~ oo. 6 Numerical results In order to solve for ~ as a function of k, we truncate the Fourier series for q(r,6~) after M terms, and use a finite-difference scheme to discretize the ordinary differential equations on a grid that has N points. We then form a compound eigenvector X of order (M+2)xN as follows: The first N positions of the eigenvector are occupied by the values of H(r) at the N discretization points ri, i = 1,2,.. N. the next N positions by the values of G(r), the next N positions by the values of q~(r), and so on; finally, the last N positions are occupied by the values of qM(r) at the discretization points. The discretized equations can then be combined into a single matrix equation of the form: A.X = mB.X (51) where A. B are compound matrices of order ((M+2) N) x ((M+2) ~). In general, depending on the value of M which is required for the convergence of the Fourier series, the order of the eigenvalue problem can become quite high, and require enormous amounts of computation (the number of operations is of order (M+2)3N3). We note however that the coefficients of the Fourier series are not directly coupled with each other, but, indirectly, through the variable G. which is of order F2. As a result, for low values of the Froude number F. which are mainly of interest here, the coupling between the coefficients qn is very weak. This, combined with the fact that for F=0 only the n=1 mode is unstable, allows an accurate representation of the Fourier series using only a low number of terms. The decomposition of the pressure field into modes I and II proves quite helpful in that respect too, since, say for mode I, truncating the series at the (2M+l)-th term requires the use of M coefficients in the Fourier series. For the finite difference grid, the domain was truncated at R = 6, and N= 80 grid points were used in a second-order finite difference scheme. At the two ends of the interval all perturbation quantities were set equal to zero. The free surface elevation A(r) can easily by determined from the computed values of H(r), which occupy the first N positions of the compound eigenvector x, as follows: P(, ) (kU(r3-~) (5V For a self-similar average flow, the following non- dimensional velocity distribution can be used: U(r) = 1 rum exp(-ar2) (53) where um, a are constants. In this paper the values um= 0.368, a= 0.89 were used. Those are the values for the self-similar profile measured by Ogata and Sato, 1966, far behind an axisymmetric body in unbounded fluid. The numerical results obtained using (53) are discussed in the next two subsections. We only discuss Mode I, since Mode II was found for this velocity profile to be stable. For velocity profiles with larger velocity deD~cits, both modes can become unstable. 558
6.1 Form of the instability waves When the Froude number is equal to zero, the Fourier coefficients become uncoupled, i.e. each one represents a different wave, and only the n=1 coefficient represents an unstable wa-ve. The eigenvalue problem is therefore identical with the one in infinite fluid, which is unstable, with the maximum amplification occuring for a wavenumber equal to 0.55, and giving a complex frequency (0.4524, 0.0172) The variation of the amplitude of the fast coefficient as a function of r is shown in figure 4a (semi-log scale). Subsequently, a finite value of the Froude number was used, using three terms in the Fourier expansion: n= 1, n=3, n=5. The series was thus truncated at the seventh term. The variation of the amplitude of the three coefficients qn as a function of r is shown in figure 4b. The first coefficient is indistinguishable from the one found in the F=0 case. The other two coefficients are extremely small; the maximum value of the n=3 coefficient is for example about 104 times smaller than the n=1. The Fourier series can accurately be represented using the n=1 term only, as further attested by the fact that the complex frequency has changed by less than one per cent from its F=0 value. As the Froude number is further increased, the importance of the other terms is gradually increased. At F=1.5 the computation was repeated using five terms (n= 1, 3, 5, 7, 9. The maximum value of the n=3 term is now about 103 times smaler than that of the first (figure tic). The complex frequency has changed somewhat, to a value (0.4526, 0.0166); the real part has thus changed by less than one per cent, whereas the imaginary part has been reduced by four per cent. This implies that increasing the Froude number has a stabilizing effect on Mode I. At F=2.5 the n=3 term is still considerably more than 10+2 times smaller than the first. The contribution of the Fourier coefficients with n ~ 1 is more visible in the far field. The complex frequency has changed to the value (0.4527, 0.0155), showing a decrease in the growth-rate of about eight per cent from the F=0 value. F=0 - o. D1 1 E-2 _ - 1 E 3 _ 1 E-4 _ l E S _ - , 1 , ~. 1 , 1 , 3 4 5 6 1 E-6 0 1 2 \ Figure 4a: Variation of the amplitude of the most unstable pressure eigenmode with r for zero Froude number (semi-log F=O.S O.1 ~E-: lE-3t lE 4' 1 E-! Figure 4b: Variation of the amplitude of the Fourier coefficients n= 1,3,5 with r for the most unstable wave at F=0.5 (semi-log scale). F=1.5 E-2 _ OFT 559 n= Figure 4c: Variation of the amplitude of the Fourier coefficients n=1,3,5,7,9 with r for the most unstable wave F=1.5 (semi-log scale).
1~ o.' 1 E-; _ i I. x - - lF_1 1 E-' 1F' ,'_' r Figure 4d: Variation of the amplitude of the Fourier coefficients n=1,3,5,7,9,11,13 with r for the most unstable wave at F=2.5 (semi-log scale). The variation of the free surface displacement generated by the most unstable wave along an x = constant plane as a function of the coordinate y is shown in figure Sa for Froude number equal to 0.5. (Only the part 0 < y < so is shown). The free surface elevation basically follows the F=0 pressure eigenmode and decays exponentially with the distance y far from the region where shear is present. The phase of the free surface elevation of the same eigenmode is shown in figure Sb (again only the part 0 c y < oo is shown). The free surface elevation in the middle of the wake, where the fluid velocity is reduced, lags behind the elevation outside the wake, where the fluid velocity has its free-stream value. This is basically the variation of the phase of the pressure eigenmode for zero Froude number. As the Froude number is increased, the free surface elevation is also increased, proportionally to the square of the Froude number. The shape of the elevation changes however by very little, owing to the aforementioned separation in magnitude between the n=1 and the subsequent Fourier coefficients. It is only after F=2.5 that the latter become significant enough to start altering the shape of the elevation. This can be seen in figure 6a, where the free surface elevation divided by the square of the Froude number Is plotted as a function of r for F-0.5, 1.5,2.5. The effect of the Froude number is more visible in the phase of the free surface elevation (figure fib). n, ~ . . O.OB _ A common feature of figures 4a-4d, where the Froude number is not too high, is that there is a wide separation in 0.06 magnitude between the n=1 and the subsequent coefficients. For large n, the Fourier series converges faster than n~3; this can be seen in figure 4e, where the Fourier coefficients for F=2.5 have been multiplied by n3 and replotted as a function of r in a linear scale. In fact equations (33), (34) suggest that, with the exception of the region around r=0, the convergence is like n - . The convergence rate is therefore algebraic, and we can safely say that the main advantage in using the Fourier series expansion, as opposed to a direct numerical solution of (10), lies in the wide separation in magintude between the first and the subsequent Fourier coefficients, which allows an accurate representation of the series using very few coefficients. n rid 0.03 _ 0.0 ~E - n=?' ~:=~=3~, , . 1 , 1 , 1 , 51 2 3 4 5 6 Figure 4e: Plot of qnxn3, n= 3,5,7,9,11,13 as a function of r, for the same wave as in figure 4d, exhibiting a convergence rate faster than n~3. 560 0.04 _ O.- ~ ~\\ Figure Sa: Amplitude of the free surface elevation as a function of y, for the most unstable wave at F= 0.5. o.lS ~ O.1 _ 0.05 ~.1 -0.15 _ -0.2 , 1 , s Figure Sb: Phase of the free surface elevation as a function of y for the most unstable wave at F= 0.5.
nit. - ~ - 0.25 no 11 t' 0.1 n, 0.2 0.15 n1 _ _._ _ _ . .. .1 ~'.-', 11 '- l /'(~-~ ~,~ , 1 l 0 1 2 3 an' - 1 , 1 , ~1 4 5 6 Figure 6a: Amplitude of the free surface elevation as a function of y for F= 0.5, 1.5, 2.5. . _ .... .. _ ... _ , 1 , 1_ , 1 2 (I 41h . and; 11 t 1 -0.2 Finally, in the infinite Froude number limit there is a symmetric mode that becomes unstable. More specifically, when F = on, the boundary condition on the free surface becomes p=O. Then an exact solution of the problem is given by: p(r,0) = q 1 (r) sin(~) (54) where q ~ ( r) is the pressure eigenmode in the zero Froude number case. The frequency of (54) is the same as the F=O. The fact that the F= oo mode is just the F=0 mode rotated by ~/2 in the y,z plane is due to the axisymmetry of the basic flow. In a non- axisymmetric flow the two limits will be different. 0.02aCF 0.02: 3- 0.015 OF ~ , 1 , 3 y play 4 5 6 Figure fib: Phase of the free surface elevation as a function of y for F= 0.5, 1.5, 2.5. As discussed above, the growth-rate of the unstable waves is consistently decreased as the Froude number is increased, showing that the latter has a stabilizing effect. This can be seen in figure 7, where the whole unstable frequency range has been plotted for three different values of the Froude number, F= 0, 0.5, 1.5. The cut-off frequency, marking transition from stable to unstable waves is reduced as the Froude number is increased, and the growth-rates of the unstable waves are also decreased. Increasing the Froude number causes therefore a, slow, but clear, "shrinking" of the unstable wave range. At the cutoff frequency, the frequencies of the growing and the decaying mode coalesce. Consequently, the condition OD/3co = 0 is satisfied there, where D ( in, k ;F ) = 0 denotes the dispersion relation of the flow at Froude number F. This condition is typical of the onset of instability in non- dissipative media. a_ F=0.5 i, l~,\ "\ __-3 _ / ' . o {,,,_ , __1, , 1 ,1~* n t`oC ^= 075 0.25 0.5 lo, arias = 0 g ~, I Figure 7: Growth-rate as a function of frequency in the unstable wave range, for F=0,0.5,1.5. (Note, that at the "cut-off' point aD/a~=o). 6.2 Spatial instability The physical character of the wake instability was determined next, i.e. whether it is of the absolute or of the convective type. As mentioned before, if mapping of the k-real axis through the dispersion relation into the cl)-plane yields frequencies with positive imaginary parts, the flow is unstable. This was done in the previous subsection, where an unstable wavenumber range was found, by solving the dispersion relation with respect to the frequency. In order to distinguish whether this instability is absolute, or convective, the "pinching" double roots (Bers, 1983) of the dispersion relation have to be determined. More specifically, one has to determine a complex pair (mo,ko) that satisfies: D(mo,ko;F) = ak (C)o~ko;F) = 0 (55) plus the "pinching" requirement, stating that the double root should be formed from the coalescing of a right-travelling with a left- travelling wave. Then it can be shown (Bers, 1983) that, if the imaginary part of me is positive, any localized perturbation leads to 561
a growing motion at any location in space, and the instability is 0-03 termed absolute; if the imaginary part of NO is negative, any localized perturbation will be convected away in finite time, leading to decaying motions at any fixed location in space; the instability is then termed convective. It can be further shown that, if the instability is absolute, the long-time response of the flow develops a normal-mode with a characteristic frequency and wavenumber. After non-linearities saturate the growth of the linear instability, the flow settles into a self-sustained oscillation at approximately this frequency and wavenumber. Absolutely unstable media are for this reason called "oscillators". Convective instabilities are on the other hand very receptive to persistent external excitations: There exists a range of excitation frequencies which lead to asymptotic states that are oscillatory in time and grow in space along the direction of propagation of the instability waves. Convectively unstable flows are thus characterized as "amplifiers". This distinction is fundamental in discussing the spontaneous appearance of unsteady patterns in viscous flows. The double roots were determined using the procedure suggested in Triantafyllou et al., 1986, and Triantafyllou et al., 1987, in which the complex wavenumber plane is mapped through the dispersion relation into the complex frequency plane; the pinch point type of double roots are located from the local angle-doubling property of the map. This procedure is particularity appropriate for the present problem, where it is possible to solve for co as a function of k, in the manner described before, but in general it is extremely difficult to do it the other way around. That the so determined double roots are of the "pinch-point" type can be verified from the fact that there is only one unstable branch of the m(k) function (Triantafyllou et al., 1987). For the velocity profile (53) the instability was found to be convective at all Froude numbers. This is due to the fact that the instability is convective when the Froude number is equal to zero, which is the most unstable case. The presence of flow reversal is required to produce an absolute instability in the self-similar wake, but such a velocity profile can not exist far behind the ship. We note that, if such a profile were present, it would remain absolutely unstable for low Froude numbers. Since the flow is connectively unstable, the question of spatial instability to an excitation periodic in time is very important. The spatially unstable waves for this problem can be determined using an iterative procedure: A complex wavenumber is assumed, the frequency is determined by solving the eigenvalue problem as explained in the previous section, and then a new guess is made untill a wave with complex wavenumber and real frequency is obtained. That the complex wavenumber corresponds to a growing, as opposed to an evanescent, mode can be verified again from the fact that only one unstable mode exists for real k. For Froude number equal to 0.5, the absolute value of the growth rate of the spatial mode as a function of frequency is shown in figure 8. (The growth-rates are actually negative, showing that the wave are amplified as x ~ on J. The most amplified spatial mode has complex wavenumber (0.65, -0.02217) and real frequency 0.5306. The real parts of its frequency and wavenumber differ thus somewhat from those of the most amplified temporal mode. The variation of the amplitudes of the Fourier coefficients of the spatially unstable mode as a function of r is shown in figure 9. The amplitude and phase of the free surface elevation are shown in figures lea, and 10b respectively. The more clear difference between the spatially unstable mode and the temporal modes discussed before can be seen in the variation of the phase with the distance from the wake axis. Figure 10b shows that a phase-lag is present in the spatial mode over a larger distance than in the temporal mode, which implies that the wavecrests of the spatial mode will be more curved. 562 1 E-4 1F /~ \ 0.02 0 0.2 1 , 1 0.4 0.6 11 , 1 0.8 ~ (1), Figure 8: Absolute value of the spatial growth-rate as a function of frequency for F = 0.5. 0.1 _ 1 1 E-2 _ 1 E-3 _ - - ~1 4 56 Figure 9: Variation of the amplitude of the Fourier coefficients n = 1, 3, 5 with r for the most unstable spatial mode at F=O.S (semi- log scale). Given the convective character of the wake instability, it is the spatial modes that are the physically significant ones. It is therefore interesting to obtain a visual picture of how their free surface manifestation can be expected to look like. A perspective view of the free surface elevation caused by the most amplified spatial mode at Froude number F=0.5 is shown in figure 11. The plot has been constructed in a frame of reference that moves with the phase velocity of the wave. In this frame of reference the wave appears stationary in time, and its spatial growth can be seen more clearly. The free surface elevation thus consists of two parallel series of alternating hills and valleys the height of which increases exponentially with x. The wave grows indefinitely according to linear theory, but in reality the growth will be eventually saturated by non-linear effects, unless the free surface wave breaks before that.
0.1 odors o.os 0.02s o t, 1 7 ~ 4 ~ ~ = _ Figure 10a: Amplitude of the free surface elevation as a function of y, for the most unstable spatial mode at F=0.5. 0.2 O.' 0.05 .05 -0.15 4~.2 J_. _ 1 I I __l__l 01 2 3 y _, 1_ 1 1 1 , _ 4 5 Figure 10b: Phase of the free surface elevation as a function of y, for the most unstable spatial mode at F=0.5. Finally, it should be mentioned that we have investigated the presence and form of spatially unstable modes in the wake of the ship, without discussing the mechanics of excitation of these modes by ambient waves. This is a subject that little is known about, and will probably attract attention in the near future. A related problem is the response of the wake to excitations in the stable wavenumber range: An incident harmonic wave will be then partly reflected and partly transmitted when it meets the region of shear flow, whereas a stable wave will be excited above the wake. It will be interesting to determine the reflected and transmitted waves in relation to the energy carried by the wave propagating above the shear flow. A study of the stable response of the wake to water waves will complement the results presented here. 7 Conclusions The instability of the wake behind a ship has been investigated numerically. The main outcome of this investigation is that the unstable waves are antisymmetric about the centerplane of the wake. The free surface manifestation of the instability wave develops a pattern consisting of two parallel series of alternating "hills" and "valleys". An interesting observation is that, at low Froude numbers, the frequency and phase velocity of the instability waves is controlled by the characteristics of the shear flow in the wake, and is practically the same as in the F=0 case. This allows the derivation of a very simple approximation for the unstable modes, as follows: We first solve for the eigenvalue co and eigenvector q'(r) for F=O; this requires the numerical solution of only an NxN eigenvalue problem. Then, for low F. the unstable eigenmode of the same flow with a free surface can be approximated by the first term of the Fourier series: p(r,~) = at (r)cos(~) + F2r(kU-~)2q~.(r) 1-(~/2)F2r(kU-~)2 2 (56) Equation (56) satisfies exactly the linearized boundary conditions at the free surface, (as in fact will any truncated series for q in (28)), and approximately the basic partial differential equation (10). Equation (56) becomes asymptotically exact as the Froude number tends to zero. The fact that the instability of the wake is convective, implies that unstable wavepackets are convected with the mean flow, and the flow can not be self-excited. Even a connectively unstable flow can, however, be driven by background noise, to generate a spatially growing response. For the wake/free surface interaction problem, a source of persistent "noise" is provided by the ambient waves, which are almost always present in the ocean. Thus, if the frequencies of the ambient waves cover the range of spatial instability, they can excite waves that grow in the streamwise direction. The vortices that are formed by this noise-driven instability can cause a local mixing in that part of the wake where shear exists, creating a region of fluid with somewhat different properties than the surrounding fluid (e.g. lower temperature, since cooler fluid from below has been brought upwards). This region, which has a width roughly equal to the width of the wake, can therefore become visible to scientific instruments and the human eye. The formation of such a region might account for the observation that ship wakes remain visible long after the passage of the ship, when all hydrodynamic disturbances have presumably been dissipated. ACKNOWLEDGEMENTS Most computations were performed using the Cray-2 of the MITSF. This work is supported by the Office of Naval Research, under Contracts N00014-87-K-0356 and N00014-90-J1312, and the National Oceanic and Atmospheric Administration, under Sea- Grant Contract NA86AA-D-SG089. 8 References 1. Batchelor, G.K. and Gill, A.E., 1962, J. Fluid Mech., 14, p. 529. 2. Benney, D.J., and Chow, K., 1986, Stud. Appl. Math., 3. Bernal, L.P., and Kwon, J.T., 1989, Phys. Fluids A, 1, p.449. 563
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DISCUSSION John P. McHugh The University of New Hampshire, USA I have recently considered very similar problems as in this paper, and I have a comment and a question. The comment is that the reduction of the quadratic eigenvalue problem to a linear eigenvalue problem may be accomplished another way. Instead of your analytical version (Eq. 38, p. 55), you can create a matrix version of the quadratic problem: [A) + B.) + C](y) = 0. Then use Z = Ay in the first two terms. The question concerns neutral modes. Did you find any neutral modes where the wavespeed is outside the range of velocity of the primary hull? I have found such waves in similar cases. AUTHORS' REPLY First, regarding the comment, the proposed alternative linearization of the eigenvalue problem seems equivalent with what I have done. I believe that the treatment I have presented is closer to the physics of the problem, but the final choice is probably a matter of preference. I am not familiar with the work of the discusser, and as he gives no specific reference for his work, I cannot make a more detailed comparison. Regarding the question, I have determined the neutrally stable modes of this problem for various Froude numbers by considering the limit of the unstable modes as their growth-rate goes to zero (see Figure 7). The phase velocities of these modes are inside the range of velocity of the basic flow. In fact, from Howard's theorem, it is straightforward to see that in the limit as the growth- rate of the unstable wave tends to zero the phase velocity has to remain within the range of the basic flow. Consequently, I do not see how in the similar problem that the discusser has considered that the phase velocity of neutrally stable modes can possibly lie outside this range. I would like to thank Prof. McHugh for his interest and comments. DISCUSSION Ali H. Nayfeh Virginia Polytechnic Institute and State University, USA The results presented in the paper show a weak influence of the Froude number on the growth-rates and bandwidth of unstable disturbances, contrary to the known results about the influence of the Mach number on the stability of compressible boundary layer. Are these results due to neglecting the influence of the Froude number on the mean-flow profile U(y,z)? AUTHORS' REPLY At. Increase of the Froude number clearly results in a reduction of the growth-rates and of the unstable wavenumber range, and, in the infinite Froude number limit, Mode I gets stabilized. Overall, therefore, the influence of the Froude number cannot be considered weak. The influence is weak only for low Froude numbers, a result that simply reflects the physics of the problem: at low Froude numbers, the presence of the free surface reduces the motion of the fluid in the vertical direction. As a result, the motion of the fluid is confined mostly in the horizontal direction, and the free surface is merely deformed to accommodate the non-uniform pressure caused by the fluid motions. This behavior does not change until the Froude number becomes high enough for the free surface to behave like an almost perfectly compliant boundary, and accounts for the weak influence of the Froude number on the instability mode. I believe, therefore, for low Froude numbers, the weak influence of the Froude number on the instability is a realistic result. I do agree, however, that, above some value of the Froude number, the average flow itself might start to change significantly in a manner that would accelerate the stabilization. There are no data available, to my knowledge, as to what that value may be; existing steady computations of the wake behind a ship (Ref. 22) show a velocity profile similar to the one used in this paper. I would like to thank Prof. Nayfeh for his interest and comments. 565
