Experiments in Fluids 32 (2002) 242±251 Ó Springer-Verlag 2002 DOI 10.1007/s003480100354

The rule of affine similitude for the force coefficients of a body oscillating in a uniformly stratified fluid E. V. Ermanyuk

242

Abstract This paper presents a study on af®ne similitude for the force coef®cients of an arbitrary body oscillating in a uniformly strati®ed ¯uid. A simple formula is derived that gives a relation between the force coef®cients for a body oscillating in homogeneous and uniformly strati®ed ideal ¯uids. In particular, it implies the existence of a universal nondimensional similitude criterion for a family of af®nely similar bodies, namely, the bodies that can be transformed into each other by vertical dilation of the initial coordinate system. Theoretical results are veri®ed by experiments with a set of spheroids having different length-to-diameter ratios. The experimental technique for evaluation of the frequency-dependent force coef®cients is based on Fourier analysis of the time-history of damped oscillation tests. Keywords Af®ne similitude, Strati®cation, Internal waves, Added mass, Damping

1 Introduction Experimental hydrodynamics is based on the theory of dynamic similitude, which yields basic criteria such as the Reynolds number and the Mach number. In addition, for certain hydrodynamic problems, one can effectively use the notion of af®ne similitude. A classic result based on the idea of af®ne similitude is the Prandtl±Glauert formula, which gives the relation between the lift force on a wing in Received: 25 September 2000 / Accepted: 6 July 2001 Published online: 29 November 2001

E. V. Ermanyuk (&) Lavrentyev Institute of Hydrodynamics, Siberian Division of Russian Academy of Science, Novosibirsk 630090, Russia e-mail: [email protected] Tel.: +7-3832-332553, Fax: +7-3832-331612 This study was supported by the Siberian Division of the Russian Academy of Sciences (grant 6 for young scientists; grant 1±2000 of Integration Project) and by the Russian Foundation of Basic Research (grant 0001±00812). The author wishes to thank Prof. A.A. Korobkin and Prof. I.V. Sturova for their interest in this study and for stimulating discussions. The author is grateful to the reviewers for valuable suggestions and comments as well as for drawing the author's attention to some important references. Prof. N.I. Makarenko and Prof. S.V. Sukhinin are thanked for the discussions concerning the revision of the paper. Thanks are due to Dr. N.V. Gavrilov for his help in experiments. The author is grateful to Dr. B. Voisin for interesting scienti®c correspondence.

a subsonic ¯ow of compressible and incompressible ¯uids (see, for example, Robinson and Laurmann 1956; Landau and Lifshitz 1959; Loitsyanskii 1970). The Prandtl±Glauert formula was derived by applying af®ne transformation of the coordinate system so that the governing equation of the thin-wing theory was reduced to a Laplace equation. Correspondingly, the effect of compressibility was shown to be equivalent to a certain distortion of the wing shape in a ¯ow of incompressible ¯uid. It should be noted that af®ne transformation of coordinates allowing a simpli®cation of the governing equations has been widely used in dynamics of strati®ed ¯uids by many authors: Krishna (1968), Grimshaw (1969), and Hurley (1972, 1997), to name a few. As regards hydrodynamic loading, Vladimirov and Il'in (1991) have considered the case of nonoscillatory slow motion of a body in uniformly strati®ed ¯uid (the velocity of a body is assumed to be an exponential function of time). They show that this problem can be reduced to the problem of the motion of ®cticious af®nely similar body in homogeneous ideal ¯uid. The primary attention is paid to the limiting case of translational motion with uniform velocity at very low Froude number. As compared to the elliptic problem considered in Vladimirov and Il'in (1991), for harmonic oscillations of a body in a strati®ed ¯uid the problem may be either elliptic or hyperbolic, depending on the ratio between the frequency of oscillations and the Brunt± VaÈisaÈlaÈ (buoyancy) frequency. In the known solutions for hydrodynamic loads acting on oscillating bodies (Lai and Lee 1981; Hurley 1997; Voisin 2001), advantage is taken of the particular body geometry. For this reason, the idea of af®ne similitude, which is applicable for a body of arbitrary shape, has not received due attention. In the present study, we consider the hydrodynamic loads on a body oscillating in a uniformly strati®ed ¯uid with constant Brunt±Vaisala frequency from the viewpoint of af®ne similitude. The hydrodynamic problem for inviscid uniformly strati®ed ¯uid within the Boussinesq approximation is formulated in terms of `internal potential' (Gorodtsov and Teodorovich 1980; Hart 1981; Voisin 1991, 1994). For frequencies of oscillations higher than the Brunt±Vaisala frequency, the af®ne transformation of the initial coordinate system allows one to reformulate the problem in terms of the Laplace equation, which makes it possible to use the classic concept of added mass described in Lamb (1924), Birkhoff (1960), Kochin et al. (1963), Newman (1977). A simple formula is obtained that gives the relation between the matrix of the added mass coef®cients for a body in a uniformly strati®ed ¯uid and the

matrix of the added mass coef®cients for a body compressed in the vertical direction in an ideal homogeneous ¯uid. Using analytic continuation (Hurley 1972; Lai and Lee 1981; Hurley 1997), the relations for the matrix of the added mass coef®cients obtained for the frequency of oscillations higher than the Brunt±Vaisala frequency are extended to the case where the frequency of oscillations is lower than the Brunt±Vaisala frequency. The known solutions for hydrodynamic loads (Lai and Lee 1981; Hurley 1997; Voisin 2001) are shown to be particular cases of the proposed rule of af®ne similitude. The rule is veri®ed in the experimental part of the paper by using the technique described in Ermanyuk (2000) in application to a family of bodies (sphere, oblate and prolate spheroids with vertical axes of revolution), which are similar in the sense of af®ne transformation with dilation of the initial coordinate system in the vertical direction. The physical difference between the 2D case studied in Ermanyuk (2000) and the 3D case studied in the present paper is discussed. A version of the classic Froude method (see Birkhoff 1960; Newman 1977) is used to separate the effects of wave and viscous damping. Finally, we discuss applications of the rule of af®ne similitude for generalization and universal representation of experimental data obtained in individual experiments in uniformly strati®ed and homogeneous ¯uids.

the condition of impermeability, which can be written as u
rF ˆ v
rF, where v …x1 ; x2 ; x3 ; t † ˆ V…x1 ; x2 ; x3 †eixt is the vector of the velocity of the body. The harmonic motion of the body assumes that we can separate the timedependent part of the internal potential, pressure and velocity in the form

u…x1 ; x2 ; x3 ; t† ˆ U…x1 ; x2 ; x3 †eixt ; ixt

p…x1 ; x2 ; x3 ; t† ˆ P…x1 ; x2 ; x3 †e

;

u …x1 ; x2 ; x3 ; t† ˆ U…x1 ; x2 ; x3 †eixt :

…4† …5† …6†

Substituting (4, 5, 6) in (1, 2, 3), we obtain


x2 r2 ‡ N 2 r2h U ˆ 0
U ˆ x2 r ‡ N 2 rh U
P ˆ q0 ix N 2 x2 U :

…7†

Finally, introducing the potential U…1† ˆ U…N 2 x2 †, after some obvious manipulations, we can formulate the following problem.

2.2 Elliptic problem

2.2.1 Problem 1 The equation of ¯uid motion is  2  o o2 1 o2 2.1 U…1† ˆ 0 ; ‡ ‡ …8† o x21 o x22 a2 o x23 The equations of fluid motion p Let us consider the problem of harmonic oscillations where a ˆ X2 1=X, with X ˆ x=N. Equation (8) may of a body in an otherwise still, inviscid, incompressible be of elliptic or hyperbolic type depending on the sign of Boussinesq ¯uid. The ¯uid is assumed to be uniformly a2. First, we consider the elliptic problem, that is, the case strati®ed, i.e., the buoyancy frequency of W > 1 (a real). For the sake of convenience, let us assign  p Nˆ gdq=qdx3 ˆ const, where q…x3 † is the density the superscript (1) to other variables of Problem 1 so that   distribution over the vertical coordinate x3 of the Cartesian o o 1 o …1† coordinate system (x1, x2, x3) and g is the gravity accel- U ˆ U…1† ; ; ; ox1 ox2 a2 ox3 eration. The equation of ¯uid motion can be written in terms of internal potential u(x1, x2, x3, t) as follows P…1† ˆ q0 ixU…1† : (Gorodtsov and Tedorovich 1980; Hart 1981; Voisin 1991, The boundary condition on the body surface S…1† ˆ S is 1994):  2  U…1†
r…1† F …1† ˆ V…1†
r…1† F …1† ; …9† o 2 2 2 u ˆ 0 ; …1† r ‡ N r h 2 ot where r…1† ˆ r, F …1† ˆ F. The components of the force vector y…1† …t† ˆ Y…1† eixt acting on the body can be obwhere r ˆ …o=ox1 ; o=ox2 ; o=ox3 †; rh ˆ …o=ox1 ; o=ox2 ; 0†. tained by integrating the pressure over the body surface: The internal potential is a generalization of the idea of ZZ classic potential for a strati®ed ¯uid where vorticity can be …1† …1† U…1† nj dS…1† ; created by the strati®cation. The expressions for the ¯uid Yj ˆ q0 ix velocity u …x1 ; x2 ; x3 ; t† and perturbation pressure S…1† …1† p…x1 ; x2 ; x3 ; t † are where nj (j ˆ 1, 2, 3) are the components of the unit  2  inward normal vector n…1† ˆ r…1† F …1† =jr…1† F …1† j at the o 2 u …2† uˆ r ‡ N r h body surface S(1). Now, let us introduce the af®ne transot2 formation of the coordinate system  2  o o 2 …10† ‡N p ˆ q0 u ; …3† ni ˆ ai xi ; o t2 ot where, depending on the index i ˆ 1, 2, 3, the coef®cients where q0 is the undisturbed density. On the body surface S of the transformation are de®ned by the function F …x1 ; x2 ; x3 † ˆ 0 (function F is …11† positive inside S and negative outside) we should impose ai ˆ …1; 1; a† 2 Theoretical background

243

Under compression of the vertical scale of the coordinate The components of the force acting on the body in Probsystem (a £ 1) in accordance with (10, 11), Problem 1 lem 1 can be expressed as transforms into Problem 2. The transformed values are …1† 3 X …1† …1† dvi denoted by the superscript (2). y ˆ m

244

j

2.2.2 Problem 2 The equation of ¯uid motion is: ! o2 o2 o2 …2† ˆ0 2‡ 2‡ 2 U o n1 o n2 o n3

…12†

iˆ1

ij

dt

Thus, our task is to ®nd the relation between the …1† …2† components of the added mass tensors mij and mij . Once this relation is known, the solution of Problem 2 can be generalized to the case of a uniformly strati®ed ¯uid. The components of the added mass tensor in Problem 2 are given by

The boundary condition that should be satis®ed on the transformed body surface S(2) described by the function ZZ …2† …2† …2† F …2† …n1 ; n2 ; n3 † ˆ 0, which is positive inside S(2) and negmij ˆ q0 Ui nj dS…2† ative outside, looks as follows

r…2† U…2†
r…2† F …2† ˆ V…2†
r…2† F …2† ;

…16†

S…2†

…13†

The relation between Problem 2 and Problem 1 can be where r…2† ˆ …o=on1 ; o=on2 ; o=on3 † and the components of easily obtained by transforming the integral (16) over the vector of the body velocity V…2† undergo the same af- body surface into an integral over ¯uid volume using the ®ne transformation as the coordinates ni , so that Gauss±Ostrogradskii formula. To do this, we surround the body surface S…2† by the control surface R…2† and consider …2† …1† Vi ˆ ai Vi : the integral over the combined surface In principle, we can consider the oscillations of a body in a Z Z …2† …2† ¯uid domain bounded by an external impermeable mateUi nj dS…2† : rial surface. The boundary conditions at such a surface S…2† ‡R…2† and the conditions at in®nity will be discussed below. The components of the hydrodynamic force acting on This integral can be expressed as an integral over ¯uid volume H…2† comprised between S…2† and R…2† in the form the body can be obtained by integrating the pressure …2† …2† …2† ZZZ Z Z P ˆ q0 ixU over the body surface S …2† …2† Yj

ZZ

ˆ

q0 ix

…2† …2†

…2† U…2† nj dS…2†

Ui nj dS…2† ˆ

;

S…2† …2† where each nj is a component of the unit inward normal vector n…2† ˆ r…2† F …2† =jr…2† F …2† j at the body surface S…2† .

S…2† ‡R…2†

H…2†

oUi dH…2† : onj

…17†

Note that (17) requires the outward unit normal at R…2† so that for the ¯uid volume H…2† the normal vectors at S…2† and R…2† are directed outward. In principle, we can think about R…2† as a ®xed material surface and generalize the statement of the problem. To do this, we should complete Problem 2 by imposing the impermeability condition at …2† R…2† described by the equation FR …n1 ; n2 ; n3 † ˆ 0; which can be written as

As is apparent from (12) and the boundary condition (13), Problem 2 is simply the classic problem of the oscillations of a body in an ideal homogeneous ¯uid. The general approach to this problem, which is based on the concept of added masses, is thoroughly described, for example, in Lamb (1924), Birkhoff (1960), Kochin et al. …2† (1963) and Newman (1977). Following Newman (1977), let r…2† U…2†
r…2† FR ˆ 0 : …18† us express the components of the force acting on the body This condition corresponds to the impermeability condiin Problem 2 in the form tion in Problem 1 imposed on the surface R…1† described by …1† …2† 3 X …2† dv the equation FR …x1 ; x2 ; x3 † ˆ 0 …2† i

yj

ˆ

iˆ1

mij

dt

;

…1†

U…1†
r…1† FR ˆ 0 :

where the components of the instantaneous body velocity …2† …2† …2† are vi ˆ Vi eixt . The components mij of the added mass tensor can be determined from the solution of Problem 2 …2† for unit potentials Ui introduced by the decomposition …2†

U

ˆ

3 X iˆ1

…2†

…2†

Vi Ui

:

…14†

Similar decomposition is assumed for Problem 1:

U…1† ˆ

3 X iˆ1

…1†

…1†

Vi Ui

:

…15†

…19†

As discussed, for example, in Newman (1977), the integrals over combined surface S…2† ‡ R…2† can be considered separately (actually, the body surface and the control surface are initially supposed to be connected by a tube of in®nitely small radius). Obviously, the conversion factors due to transformation to the initial coordinate system of Problem 1 for the left-hand and right-hand sides of (17) must be the same. The derivation of these conversion factors for the integral over H…2† is straightforward. Keeping in mind (14) and (15), and introducing the tensors of the nondimensional added mass coef®cients …1† …1†  …2† …2†  Kij ˆ mij q0 W …1† and Kij ˆ mij q0 W …2† , where W …1†

and W …2† are the volumes of bodies surrounded by S…1† and the functions fij depend on a number of additional arguS…2† , we can formulate the following relation: ments describing the geometry of R…1† . In the elliptic problem R…1† may be a closed material surface. …1† …2†

Kij

…2† Kij

ˆ

Vi nj …1† V xj i

Note that W …2† ˆ aW …1† . Finally, making use of the coef®cients of the af®ne transformation ai from (10), the rule of af®ne similitude for the added mass coef®cients in Problem 1 (8), (9), (19) and Problem 2 (12), (13), (18) is expressed by the following simple formula …1†

…2†

Kij ˆ Kij ai aj :

…20† …2†

When the surface R…2† tends to in®nity, the value Kij tends to the value for an isolated body submerged in in®nite ¯uid volume. It is well known (Milne-Thomson …2† 1960) that if Kij in an in®nite ¯uid is of order O(1), then the leading term of the additive correction due to the presence of material surface R…2† is of order O…j3 † and O…j2 † in 3D and 2D problems, respectively, where j is the ratio between the characteristic length of the body and the distance between the body and the surface R…2† . A detailed discussion on the far-®eld behaviour of the potential U…2† can be found in Newman (1977). In practice, once we are concerned with evaluation of the force on a body in a uniformly strati®ed ¯uid of in®nite extent, the most important requirement at W > 1 (elliptic problem) is that the volume of the test tank should be large compared to the volume of the body. Below, we shall discuss the additional requirements to the control surface (equivalently, to the test tank) and the conditions at in®nity that should be imposed in the case W < 1 (hyperbolic problem). …1† In the ¯uid of in®nite extent, the properties of Kij and …2† Kij depend only on geometry of the oscillating body. In particular, when the body has three planes of symmetry x1 ˆ 0, x2 ˆ 0 and x3 ˆ 0, (20) reduces to …1†

…2†

…1†

…2†

…1†

…2†

K11 ˆ K11 ; K22 ˆ K22 ; K33 ˆ K33 a2 ; while the nondiagonal coef®cients are identically zero (Lamb 1924; Newman 1977). Let us consider a body with characteristic dimensions b1, b2 and b3 along the directions x1 , x2 and x3 , respectively. For a body oscillating in ideal homogeneous ¯uid of in®nite extent, the analytic solutions of Problem 2 are usually represented in the form of functional dependence …2†

Kij ˆ fij …e; q† ;

…21†

where e ˆ b2 =b1 and q ˆ b3 =b1 . The af®ne transformation (10), (11) affects only the ratio q while the ratio e remains constant. Once the dependencies fij …e; q† are known, the added mass coef®cients of a body with given e0 and q0 oscillating in a uniformly strati®ed ¯uid with frequency W can be readily obtained in accordance with  (20) by the p simple substitution q ˆ q0 a ˆ q0 X2 1=X in (21) as follows …1†

Kij …X† ˆ fij …e0 ; q0 a†ai aj

…22†

Similar functional relationships can be written when the effects due to R…1† are taken into account. In the latter case,

2.3 Hyperbolic problem Now, let us discuss the solution of the hyperbolic problem. As discussed in Hurley (1972), Lai and Lee (1981) and Hurley (1997), the solution for W > 1 (elliptic problem) can be extended to the case W < 1 (hyperbolic problem) by means of analytic continuation. This possibility follows from the fact that the solution in the frequency-domain can be obtained from the Fourier-transform of a causal function of time. A detailed analysis of the approach in terms of the causal Green's function and the literature survey can be found in Voisin (1991, 1994). Actually, in the experimental part of the paper we use the Fourier transformation of a causal function (namely, impulse response function) for derivation of the frequency-dependent force coef®cients. It should be emphasized that the condition of causality implies that the energy of internal waves generated by the oscillating body should be radiated to in®nity. Accordingly, the volume of strati®ed ¯uid around the oscillating body should be in®nite. Let us illustrate this requirement by a simple example. Consider a body of neutral buoyancy in ideal uniformly strati®ed ¯uid surrounded by a rigid closed envelope. The body is given a vertical impulse at the time instance t ˆ 0. The resulting internal waves and the response of the body r(t) persist in the future for an in®nitely long time without any decay because of the wave re¯ections. Correspondingly, the Fourier integral of the impulse response function r(t) is not converging at t ! 1. Moreover, owing to accumulating phase differences between the elementary wave disturbances travelling in different directions between the body and the envelope, the response r(t) at t ! 1 will have the features of a stochastic process so that, ultimately, the cause-and-effect relation is lost. For the above reasons, r(t) of a body surrounded by a closed material surface is not a causal function of time. Hence, the solution of the elliptic problem can be continued analytically for the hyperbolic case only when R…1† is not a closed surface and the internal wave energy is radiated to in®nity. For example, analytic continuation is possible if the body oscillates between two horizontal planes with no lateral boundaries. It is interesting to note that the above-described effects can be easily observed experimentally. If the experimental tank is not equipped with the absorbers of wave energy, one can obtain reliable measurements of the force coef®cients only for W > 1, while at W < 1 the frequencydependent values show very large scatter (an experimental analogy for a non-analytic function of frequency). In fact, the absorbers technically ful®ll the radiation condition formulated in the causal sense: waves are radiated away from the source to in®nity and never return back. It should be noted that in a real strati®ed ¯uid the internal waves generated by an oscillating body are attenuated with distance because of viscous dissipation. In that sense, the role of viscosity in experiments is similar to the role of the wave absorbers.

245

When W < 1, parameter a2 is negative (a is complex). In this p case, we can introduce the real-valued parameter  g ˆ 1 X2 =X. The discussion on the proper choice of branches for multivalued functions at W<1 is given in Hurley (1972, 1997). Accordingly, the analytic continuation for a is ig. The coef®cients ai should be replaced by the coef®cients ci ˆ …1; 1; ig†. With this notation, expression (22) for a body located in a uniformly strati®ed ¯uid of in®nite extent becomes 246

…1† Kij …X†

ˆ fij …e0 ;

q0 ig†ci cj

…23†

As emphasized in the above discussion, for a body oscillating near rigid boundaries, an analytic continuation similar to (23) can be constructed only when the wave energy is radiated to in®nity so that causality is satis®ed. This condition constitutes the fundamental difference between the hyperbolic and elliptic cases. …1† In general, at W<1 the coef®cients Kij are complexvalued. Correspondingly, Lai and Lee (1981) have presented their results by introducing the amplitude and phase of the complex added mass coef®cients. A similar concept is used in Hurley (1997). Physically, the real part …1† of Kij corresponds to the force component that oscillates in-phase with the acceleration of the body (inertial force), …1† while the imaginary part of Kij corresponds to the force component that oscillates in phase with the velocity of the body motion (damping force). Since the real and imagi…1† nary parts of Kij have different physical meanings, it …1† seems reasonable to decompose mij as is customary in marine hydrodynamics (see, for example, Wehausen 1971; Newman 1977):

kij ˆ lij i …24† x where lij and kij are commonly referred to as `added masses' and `damping coef®cients', respectively (see Wehausen 1971; Newman 1977). It should be noted that lij and kij are both real. Causality requires that kij and lij should satisfy so-called Kramers±Kronig relations (Kotik and Mangulis 1962; Landau and Lifshitz 1980) that can be used to test the physical correctness of the analytic continuation. The nondimensional quantities for lij are called `added mass coef®cients' or Ôinertia coef®cients' …1† mij

In the experimental part of the paper we will show that the behaviour of the inertial coef®cients is consistent with an inviscid scenario, while the damping coef®cients are markedly affected by viscous effects. Using lij and kij de®ned in (24), one can write the equation of the body motion in three degrees of freedom in the frequency domain (Cummins 1962; Wehausen 1971; Newman 1977) 3 h X jˆ1

 i M dij ‡ lij xj ‡ kij x_ j ‡ cij xj ˆ hi eixt

…27†

where M is the inherent (mechanical) inertia, dij is the Kroneker delta (dij ˆ 1 if i ˆ j and dij ˆ 0 otherwise), hi is the amplitude of the exciting force in direction xi, cij are the components of the matrix of restoring force coef®cients and overdot denotes differentiation with respect to time.

2.4 Similitude condition As apparent from (22) and (23), the results of measure…1† ments of Kij in a uniformly strati®ed ¯uid of in®nite extent can be expressed in such a way as to relate the function fij …e; q† at W > 1 and its analytic continuation fij …e; iq† at W < 1. In other words, the results of experiments with a family of bodies (say, spheroids) characterized by different values of q0 can be matched at a common curve. To avoid the use of complex-valued arguments for representation of physical dependencies, we can introduce
fij e; q20 a2 . For a set of af®nely similar bodies with …1† different q0, we have Kij =…ai aj † ˆ idem if q20 …X2

1†=X2 ˆ q20 a2 ˆ idem :

…28†

2.5 Particular cases To illustrate the approach described above, let us consider the solutions for an elliptic cylinder (Hurley 1997) and for a spheroid with a vertical axis of revolution (Lai and Lee 1981). For an elliptic cylinder with horizontal semi-axis b1 and vertical semi-axis b3, the functions that express the dependence of the added mass coef®cient on q ˆ b3/b1 are: …2† …2† K11 ˆ q (horizontal oscillations) and K33 ˆ 1=q (vertical oscillations). Substituting these functions into (22) and l Cij ˆ lij =q0 W ; …25† (23), one immediately obtains Hurley's (1997) solution. The expressions for the added mass coef®cients of an oblate spheroid with a vertical axis of revolution (see, for where W ˆ W …1† . The term inertia coef®cient is widely used in the literature on forces acting on a body oscillating example, Korotkin 1986) are: p p
in a viscous ¯uid, when both inertial and dissipative effects q arcsin 1 q2 q 1 q2 …2† …2† are signi®cant (see, for example, Bearman et al. 1985). p p K11 ˆ K22 ˆ …2 q2 † 1 q2 q arcsin 1 q2 The nondimensional damping coef®cients are de®ned as …29† p p Cijk ˆ kij =q0 WN : …26† 2 2 q arcsin 1 q 1 1 q …2† p p …30† K33 ˆ …1† q arcsin 1 q2 q 1 q2 Making use of the force coef®cients Kij , we can write   …1† …1† The solution by Lai and Lee (1986) for K33 can be readily Cijl ˆ Re Kij obtained from (30) by substituting q ˆ q0 a for W>1 and   …1† q ˆ q0 ig for W<1 in accordance with (22), (23). The k : Cij ˆ XIm Kij …1† formula for K11 can be derived from (29) in a similar

manner. Let us note that in the case q0 ˆ 1 (sphere), (29) yields the result that coincides with the one obtained by Voisin (2001). Voisin (2001) found a correct form of the distribution of singularities over the surface of a sphere in a uniformly strati®ed ¯uid that allows the hydrodynamic loads to be evaluated. When making a comparison with the original formulas presented in Lai and Lee (1986) and Voisin (2001), one should in mind the important  keep p
identity: arccot 1 x2 =x2 ˆ arcsin x. It should also be mentioned that, strictly speaking, (29) and (30) are applicable for an oblate spheroid, i.e., for q < 1. For q > 1 the arguments of the functions are complex-valued. Care must be taken about the proper determination of multivalued functions; standard mathematical software may not necessarily return the appropriate value. …2† Analytical formulas for Kij are known for a wide variety of geometrical shapes (Riman and Kreps 1946; Korotkin 1986; some references are given in Birkhoff 1960). The approach described in the present paper allows one to …1† use these formulas to obtain the force coef®cients Kij , i.e., to solve the problem for a uniformly strati®ed ¯uid. However, it should be kept in mind that additional theoretical study may be necessary for a correct construction of analytic continuations for the functions fij …e; q†, which may not be straightforward in some cases. In experiments described below, our attention is fo…1† cused on the force coef®cient K11 for horizontal oscillations of a spheroid with vertical axis of revolution. We present an experimental proof of the af®ne similarity for a family of spheroids with different q0. The case of spheroid geometry is chosen for relative simplicity of manufacturing. Additionally, it gives a clue to the understanding of the main physical features of the 3D problem as compared to the 2D problem studied in Ermanyuk (2000).

3 Experiments 3.1 Experimental set-up The experimental technique used in the present study is similar to the one described in Ermanyuk (2000), where an interested reader can ®nd detailed information. In what follows, we present a brief description of the experimental approach with special emphasis on those details that are different from Ermanyuk (2000). The experiments were carried out in a test tank (80 cm wide, 40 cm deep and 100 cm long). A sketch of the experimental installation is shown in Fig. 1. The test tank was ®lled with linearly strati®ed ¯uid to the depth H ˆ 0.32 m. The Brant±Vaisala frequency in all experimental runs was N ˆ 1.0 rad/s. The linearity of the density distribution was checked by the conductivity probe. A weak solution of sugar in water was used to create the density strati®cation. Let us note that the physical properties of such a solution are quantitatively and qualitatively close to the properties of the glycerine-water solution used in Ermanyuk (2000). The dynamic viscosity of the sugar-water solution varied from 1.14´10)3 kg/(m s) at the free surface, up to 1.55´10)3kg/ (m s) close to the bottom of the test tank. Here, the ®rst value corresponds to the dynamic viscosity of pure water

247 Fig. 1. Experimental installation

at the temperature T ˆ 15°C. The temperature was fairly constant (to within ‹ 1°C) throughout the experiments. Correspondingly, the variation of the dynamic viscosity with temperature did not exceed ‹ 3%. The reference density q0 in (25) and (26) is q0 ˆ 1.016 g/cm3. It is the ¯uid density at the depth H/2. The experiments were performed for a set of spheroids with vertical axes of revolution. The maximum diameter of the horizontal cross section of the spheroids was D ˆ 6.4 cm. The studied ratios between the length of the vertical axis L and the diameter D of the spheroids were L/D ˆ 0.5, 1, 2 (oblate spheroid, sphere and prolate spheroid, respectively). The centre of the spheroids was submerged to the depth 0.5H. In the experiments, the ratio H/L was suf®ciently large to be, in practical terms, considered in®nite. Otherwise, to have af®nely similar experimental systems, we should have kept the value H/L constant in all experiments. The upper part of the pendulum was equipped with a vertical micrometric screw and a nut. The variation of vertical distance between the gravity centre of the nut and the axis of pendulum rotation allowed the restoring moment of the pendulum to be changed. As a result, the frequency of damped oscillations could be varied. The restoring moment coef®cient was measured by static calibration in situ. The oscillations of the pendulum were excited by the impulse produced by dropping a steel ball on a pretensioned rubber membrane attached to the horizontal bar of the pendulum. The motion of the pendulum was measured by an electrolytic displacement sensor. The analog output of the sensor was sampled by a 12-bit A/D converter at sampling frequency 20 Hz. The time-history of damped oscillations was then analysed by an IBM personal computer. To prevent the re¯ection of internal waves from the side walls, the whole perimeter of the test tank was equipped with wave breakers.

3.2 Impulse response function analysis The analysis of the impulse response functions performed in the present paper is identical to the one described in Ermanyuk (2000). It is essentially based on the well-known fact (see, e.g., Cummins 1962) that the response of a linear system x(t) to an arbitrary force h(t) can be obtained by

248

summing up the impulse response functions of the system inertial coef®cients do not depend on the frequency of oscillations, and their mean measured value coincided r(t) in the form of the convolution integral with the theoretical one predicted by (29) to within the 1 Z accuracy of experiments. The damping coef®cient k was x…t† ˆ r…s†h…t s†ds : …31† found to be proportional to x1/2 in agreement with 11the 0 asymptotic Stokes theory (Stokes 1851; Landau and Lifshitz 1959; Wang 1968) for small amplitudes of oscillations In the particular case of harmonic exciting force and large Stokes numbers b ˆ D2x/m, where m is kinematic  ixt h…t† ˆ h e the integral (31) yields viscosity. Moreover, for the particular case of a sphere, the x…t† ˆ h eixt R…x† ; …32† value of the damping coef®cient agrees well with the thep and Lifshitz p Landau where the complex frequency response function R…x† is oretical prediction (see, for example, 2 1959), which yields k ˆ 3pD mx =2 2: q 0 de®ned as the Fourier transform of the impulse response function

Z1 R…x† ˆ

r…s†e

ixs

ds :

0

The amplitude and the phase of the complex frequency response function can be introduced as jR…x†j ˆ p  R2c …x† ‡ R2s …x† and h…x† ˆ arctan…Rs …x†=Rc …x††, R1 respectively, with Rc …x† ˆ r…s† cos xs ds and Rc …x† ˆ 1 R 0 r…s† sin xs ds.

4.2 Uniformly stratified fluid l k The dependencies C11 …X† and C11 …X† measured in the uniformly strati®ed ¯uid are shown in Fig. 2 and Fig. 3. Theoretical curves derived from (22), (23) and (29) are plotted by dash, solid and dot lines for L/D ˆ 0.5, 1, 2,

0

Since the angle Y of pendulum inclination is small, the body attached to the lower end of the pendulum undergoes nearly perfect horizontal oscillations so that Y ˆ lx1, where l is the distance between the axis of rotation of the pendulum and the centre of the oscillating body (in experiments l ˆ 60 cm). The motion of the experimental system in the frequency domain can be described by (27), which, for one degree of freedom, reduces to the following single equation

…M ‡ l11 † x1 ‡ k11 x_ 1 ‡ c11 x1 ˆ h1 eixt

…33†

For horizontal oscillations of a body attached to the pendulum, the inherent inertia of the system is M ˆ J/l2, where J is the total moment of inertia about the axis of rotation. Substituting (32) in (33), one can obtain the expressions for the frequency-dependent l11 and k11

l11 ˆ

 c11 1 x2

 jR…0†j cos…h…x†† jR…x†j

k11 ˆ

c11 jR…0†j sin…h…x†† : xjR…x†j

l Fig. 2. Inertial coef®cient C11 of spheroids versus oscillation frequency W. Dotted line, empty circles: L/D ˆ 2; solid line, black circles: L/D ˆ 1; dashed line, empty triangles: L/D ˆ 0.5 (lines ± theory, symbols ± experiment)

M

Here jR…0†j denotes the amplitude of the frequency response function at zero frequency. For a linear system, the normalization of jR…x†j by jR…0†j allows one to use Fourier transforms of the experimental impulse response functions at any value of the impulse excitation.

4 Experimental results 4.1 Homogeneous water A series of experiments in homogeneous water has been conducted for all the spheroids to obtain the background Fig. 3. Damping coef®cient Ck of spheroids versus oscillation 11 information on their force coef®cients in the absence of frequency W. Legend is the same as in Fig. 2. The results of experiments in a homogeneous ¯uid are marked by crosses any effects due to strati®cation. It was found that the

respectively. The same types of lines are used to represent the results of measurements performed in homogeneous water. In the latter case, the lines are marked by crosses. Note that for the data obtained in homogeneous ¯uid, the normalization (26) is used in a purely formal sense to facilitate the explicit quantitative comparison. As is easy to l see, the measurements of the inertial coef®cient C11 …X† are in agreement with the theoretical predictions while the k measured values of the damping coef®cient C11 …X† are systematically higher than the predicted values. The additional damping in a real strati®ed ¯uid is caused by viscous effects. k As is seen in Fig. 3, when W>1 the values of C11 measured in uniformly strati®ed and homogeneous ¯uids practically coincide. Thus, we can assume that in the studied range of parameters the viscous damping is weakly dependent on the presence of strati®cation. Furthermore, we may hypothesize that the total damping can be decomposed into wave and viscous components using a version of Froude's method (Birkhoff 1960; Newman 1977). Indeed, Froud's method for the decomposition of ship resistance is physically based on the fact that the length scales of wave and viscous phenomena are different. For the problem considered in the present paper, the thickness of the oscillatory boundary layer is of the order d  …m=x†1=2 (for a detailed discussion, see Hurley and Keady 1997). The typical length scale for internal wave phenomena in a uniformly strati®ed ¯uid is given by the width of internal wave beams, which has the same order of magnitude as the typical size (say, diameter D) of an oscillating body. Under experimental conditions, the ratio d=D ˆ b 1=2 was suf®ciently small. Therefore, the total damping coef®cient can be represented as a sum of wave k kw kv and viscous components as C11 …X† ˆ C11 ‡ C11 , where the kw kw wave damping is an unknown function C11 ˆ C11 …X† and kv the viscous damping is a known function C11 ˆ AX1=2 . The constant A can be evaluated either from experimental data obtained in homogeneous ¯uid or from the results of measurements in uniformly strati®ed ¯uid at W>1 when the wave damping is zero by de®nition. In practice, to evaluate the constant A we have used least-squares approximations of experimental points. Both methods of evaluation yield the same result. The depenkw dencies C11 …X† for oblate spheroid, sphere and prolate spheroid are shown in Fig. 4 along with the theoretical curves. The notations are the same as in Fig. 3. One can see very good quantitative agreement between theory and experimental data. Furthermore, as is discussed in Sect. 2, we can use the property of af®ne similitude and represent the results as a universal dependence for all spheroids with the vertical axis of revolution. To do this, we need to replot the data shown in Fig. 2 and Fig. 4 as functions of …1†
l the similitude criterion (28): Re K versus q20 a2 ˆ C11 11 …1†
kw 2 2 and Im K11 ˆ C11 =X versus q0 a . These dependencies are shown in Fig. 5 and Fig. 6, respectively. Note that in the second case we use the wave component of the total damping since the analysis presented in Sect. 2 is strictly valid only for an ideal uniformly strati®ed ¯uid. Being replotted as functions of q20 a2 , the theoretical curves shown in Fig. 2 and Fig. 4 are represented by a single

249

kw Fig. 4. Wave damping coef®cient C11 of spheroids versus oscillation frequency W. Legend is the same as in Fig. 2

…1†
Fig. 5. Real part of the force coef®cient Re K11 versus af®ne 2 2 similitude criterion q0 a . Solid line, universal theoretical dependence derived from (29), symbols are the same as in Fig. 2

…1†
Fig. 6. Imaginary part of the force coef®cient Im K11 versus af®ne similitude criterion q20 a2 . Solid line, universal theoretical dependence derived from (29), symbols are the same as in Fig. 2

function with real and imaginary parts shown in Fig. 5 and Fig. 6, respectively (solid line). As evident, all the data obtained in experiments with spheroids at q0 ˆ 0.5, 1, 2 fall on the same universal curve. The scatter of experimental data somewhat increases when q20 a2 ! 1, as might well be expected since the ®nite frequency range 0 < W < 1 is transformed into the
in®nite …1† range
1 < q20 a2 < 0. Note that Re K11 ! 1 and …1† 2 2 Im K11 ! 0 when q0 a ! 1. 250

5 Discussion It is interesting to compare the hydrodynamic loading in 3D and 2D problems. In accordance with Hurley (1997), the force coef®cients for a circular cylinder (2D case) in a p l k uniformly strati®ed ideal ¯uid are: C  0, C ˆ 1 X2 p 11 11 l 2 k 1=X, C11  0 at W>1. In the 3D at W < 1 and C11 ˆ X case the behaviour of the inertial and damping coef®cients at W > 1 is qualitatively similar to the 2D case. However, the dynamic effects at W < 1 are radically different. The difference can be explicitly explained in terms of (7). In the limiting case x ! 0, (7) becomes the 2D-Laplace equation with respect to horizontal coordinates x1, x2. Physically, it means that the horizontal ¯uid layers move essentially independently of each other. Correspondingly, the lowfrequency limit for the inertial coef®cient of a horizontally oscillating body can be evaluated by integrating the distribution of the inertial coef®cients of horizontal cross sections over the vertical coordinate x3. In particular, for a body with a vertical axis of revolution, it implies that l C11 ! 1 as X ! 0, which is in agreement with the experimental results seen in Fig. 2. As for the damping coef®k cient C11 , its low-frequency limit for the 3D problem is zero since the motion of ¯uid particles at X ! 0 occurs in horizontal planes, i.e., there is no transfer of kinetic energy into potential energy and vice versa. This feature of wave damping in the 3D problem is in striking contrast with the 2D problem (Hurley 1997; Ermanyuk 2000), where the damping coef®cient takes the largest value at X ! 0 because of the blocking effect. Similar effects for slow nonoscillatory motion of 2D and 3D bodies in a uniformly strati®ed ¯uid are discussed, for example, in Drazin (1961), Boyer et al. (1989), Vladimirov and Il'in (1991) and Lin et al. (1992). The rule of af®ne similitude discussed in the present paper gives a new insight into the classic problem of the added mass of a body oscillating in an ideal homogeneous ¯uid. Within the classic approach, an experimental veri®…2† cation of a theoretical dependence Kij ˆ fij …e; q† for a given type of body geometry has to be performed for a set of bodies with different values of e0 and q0. However, once the problem is reformulated in terms of the force coef®cients for a body oscillating in a uniformly strati®ed ¯uid, a single experiment at ®xed e0 and q0 gives the experimental con®rmation of the function fij(e0, q) for any q < q0. In particular, it implies that the results for a circular cylinder presented in Ermanyuk (2000) can be generalized for elliptic cylinders. On the other hand, if the functional dependence …2† Kij ˆ fij …e; q† already has an experimental con®rmation, say, by forced oscillation tests in a homogeneous ¯uid, there

is no need to perform any experiments in a uniformly strati®ed ¯uid since the values of the force coef®cient …1† K11 …X† can be readily obtained from (22) and (23). A note should be made on the veri®cation of the solution by Lai and Lee (1981) for a vertically oscillating spheroid in a uniformly strati®ed ¯uid. The analysis of the impulse response functions described in Sect. 3 of the present paper can be applied to the time-domain solution obtained by Larsen (1969) for the damped vertical oscillations of a sphere and a circular cylinder in a uniformly strati®ed ¯uid. With some manipulations, one can show that, being transformed into frequency-domain, Larsen's (1969) solution exactly coincides with the solution given by Hurley (1997) and Lai and Lee (1981). Moreover, the paper by Larsen (1969) also presents some original experiments on damped oscillations of a sphere, which con®rm the theoretical analysis. Thus, Larsen's (1969) paper can be also considered as an experimental con®rmation for the results by Lai and Lee (1981). Finally, let us note the remarkably wide range of applicability of the rule of af®ne similitude in the problem considered. In the Introduction to this paper, we have mentioned the Prandtl±Glauert formula for a thin wing in a subsonic ¯ow of compressible ¯uid. Experimental data show (Robinson and Laurmann 1956; Loitsyanskii 1970) that this formula is applicable up to the Mach number of about 0.6. In fact, the af®ne transformation used in the thin-wing theory assumes the dilation of vertical length scale. Hence, as the Mach number tends to unity, the initial assumptions of the thin-wing theory are violated. In supersonic ¯ow, the Prandtl±Glauert relation is not applicable at all, so that lift and drag are predicted by Ackeret's formulae (see, for example, Robinson and Laurmann 1956; Loitsyanskii 1970). By contrast, the experimental data obtained in the present study show that essentially the same relations are applicable both for W < 1 and W > 1 in the whole studied range of nondimensional frequency W, which formally plays the role similar to inverse Mach number. This remarkable feature is conditioned by the possibility of analytic continuation in frequency. Additionally, there are no initial limitations on the body shape and the af®ne transformation is that of vertical compression so that any ®nite body ultimately degenerates into a horizontal `pancake' at W ® 1.

6 Concluding remarks The rule of af®ne similitude, which gives a relation between the force coef®cients for the problems of body oscillations in homogeneous and uniformly strati®ed ¯uid, was derived theoretically and con®rmed experimentally. The experiments were carried out for a set of spheroids with vertical axes of revolution. It was shown that the results of experiments for a family of af®nely similar bodies can be presented in the form of universal dependence. It was also shown that the total damping of a body oscillating in a uniformly strati®ed ¯uid can be consistently decomposed into wave and viscous components by applying a version of Froude's method. The value of the inertial coef®cient was found to be essentially independent of viscous effects.

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