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The Calculations of Fluid Actions on Arbitrary Shaped Submerged Bodies Using Viscous Boundary Elements W. Price, M. Tan (Brunei University, United Kingdom) ABSTRACT A time dependent viscous boundary element method is developed to calculate the unsteady fluid forces acting on a rigid body moving in a stationary, unbounded, incompressible viscous fluid. The theoretical approach presented is analogous to a potential flow singularity distribution panel method. Here the singularity is replaced by a time dependent fundamental solution derived from a modified Oseen equation developed through an integral equation formulation involving convolution time integrations. Analytical expressions are given for the unsteady and steady state fundamental solutions appropriate to two- and three-dimensional fluid-structure interaction problems. By distributing the viscous panels over the wetted surface of the body and into the fluid domain, a numerical scheme of study is devised to solve the non-linear time dependent integral equations. To illustrate the method, preliminary results are presented of the unsteady and steady state fluid actions and flow fields associated with arbitrary shaped, submerged bodies moving through prescribed manoeuvres. INTRODUCTION When a rigid body departs from steady motion in a straight line the surrounding fluid exerts a resultant force and a resultant moment about the centre of gravity of the body as a consequence of the disturbance. In manoeuvring and control studies the variations of fluid actions to disturbances (i.e. displacement, velocity, acceleration), referred to as slow motion derivati~ OCR for page 801
In these steady flow studies, the analytically defined fundamental solution adopted to describe the viscous av 2 . element was derived from a modified Oseen's equation. at + (V.V jV = V V - Vp + f + u (1) This was chosen because Oseen's equation and Navier-Stokes' equation are qualitatively similar, so that solutions of the former are expected to yield qualitative information about solutions of Navier-Stokes equation for all Reynolds number. In reality, however, steady flow around a body exists only in the small Reynolds number flow regimes and it becomes academic (though remaining of great interest) to extend the investigation into the high Reynolds number flows where unsteady influences dominate. The previous investigations showed that by distributing viscous boundary panels over the wetted body's surface only an Oseen flow-linearised model was created, whereas distributing panels over the body's surface and into the wake a non-linear convective model resulted. The method could be readily applied to describe the forces acting on bodies of arbitrary shape, the flow field around such bodies and it was suitable to use in multi-body structure - fluid interaction problems, e.g. the flow around clusters of cylinders such as occur in a cross-section of a leg of a lattice jack-up structure. Further, after checks on convergence of solution, panel distribution idealizations etc. of the numerical procedures, in low Reynolds number flows Re < 100 theoretical predictions of the fluid actions and experimental datat14,15] showed good agreement and for higher Reynolds number flows (Re ~ 103) the results confirm the qualitative and quantitative trends observed by othersE16,171. In comparison with the non-linear model, the linear model overestimates the values of the fluid actions. The global solutions (i.e. pressure distribution, fluid actions, etc) derived by the non-linear model are relatively insensitive to panel distribution, idealization etc., allowing simplifications in the numerical procedures; however, the evaluation of a detailed flow velocity field is more sensitive to idealization and the wake fluid domain must be modelled to allow for the complete formation of the vortex wake pattern. Building on these previous studies, here we present a viscous boundary element panel distribution method to evaluate the unsteady fluid actions experienced by two and three-dimensional arbitrary shaped bodies moving in a stationary, unbounded, incompressible viscous fluid. The final objective of this study is to describe the fluid actions on a rigid body undergoing a prescribed, though arbitrary, manoeuvre, e.g. a sequence of commands involving an acceleration, constant forward speed, a deceleration etc., rather than an oscillatory motion or steady state condition. For this reason the problem is described in a body fixed frame of reference and an integral equation formulation derived involving convolution time integrations. The analytically defined time dependent fundamental solution or oseenlet adopted to describe the viscous element is derived from a modified time dependent Oseen's equation. The numerical scheme of studies devised previously for the steady state problem are adapted for this unsteady problem and preliminary findings are presented of the fluid actions and flow fields associated with a two-dimensional circular cylinder moving through prescribed manoeuvres. EQUATIONS OF MOTION In a body fixed coordinate system translating with velocity -unto, [u(0) = 0], the flow of an incompressible fluid of constant viscosity described by Navier-Stokes' equations, expressed in terms of non-dimensional Oseen variables, are given in the form 802 V.V = 0 = div V (2) where V, p, f represent non-dimensional expressions for the fluid flow velocity, pressure in the fluid and external or body force respectively. The corresponding dimensional quantities, denoted by a prime, are V'=UV U'=UU p'=p'U p, P=(Re U /L'jf, = (Lt/Re) A, t'= (L'/Re U)t, V'= (Re/L'jV, a/0t' = (Re UlL')a/0t , where L' and U denote characteristic length and velocity parameters respectively, p' represents the fluid density, A', ~ denote spatial variables and the Reynolds number Re = U L'/v' where lo' represents the kinematic viscous coefficient. Letting V = U ~ v, where U denotes a constant velocity, it follows from equations (1) and (2) that the pressure p and velocity v satisfy the equations v + (U+v).Vv = V v - Vp + f + u V.v =0 (3) where an overact denotes differentiation with respect to time. This equation of motion is non-linear, but by invoking the usual assumptions, Oseen's equation is obtained in the form v+ U.Vv=V2v-Vp+f+u (4) V.v =0 subject to the initial time condition v (t,0) = - U. INTEGRAL EQUATION FORMULATION (5) Before transforming the Navier-Stokes' equations of motion described in equation (3) into an integral equation using Gaussian integral formulae, let us introduce the linear, time dependent Oseen matrix operator e, ~(a/at+u.v-v2~-vv. Vl (6) V. 0] and its linear adjoins time dependent operator '9*_ ~ ~ (a/at - A. v -v2) - TV. _v ~ `7' -V. O Here I denotes the unit matrix, U = (U. V, W) and these operators are applicable to either two- or three-dimensional problems. Continuing this use of matrix and tensor notation we may define V [v] F ~f+u-V.(vv)

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Vs =| ps |, Fs =l 0s ] (8) and the components Plj = -P6lj+ Vl,j + Vj, PSlj PS51j + VS1J + Vsj,1 ~ In these expressions vl j = avl/axj, v*sl j = av*Sl/ax, etc. and the subscripts s, l, j, etc. take values 1,2 in a two-dimensional problem and values 1,2,3 in a three-dimensional problem. This notation allows the Navier-Stokes' equations to be written as 49 . V = F (10) in which the non-linear convective term is included in F. By introducing a convolution integration over the time domain 0~ < ~ < t and using Gaussian integral formulae we find that a Green's function identity can be constructed. Thatis ; do j{V (x,t-~.~. Vex, ~) - Vs (x,t-~. ~.V(x,~ldQ o- Q _~{VT*~*.V - V T* O.V)dQ Q where = | {Vsj * (Rj - U.n Vj) - Vj * Rsj } dE - | {vsj~xto-) vj~x,t) + Uj vsj~x,t) } dQ (11) Q R. =p n.=~-p8 +v +v ~n. (12) ~ 13 1 1] 1,] ],1 1 denotes the force component in the jth direction and * * R . = p .. n. S] SI] 1 (13) where ni is redefined as the ith component of the unit normal vector at the boundary surface ~ pointing into the fluid domain Q. In equation (11), the surface encloses the domain Q. the superscript T denotes a transposed matrix, a summation convention holds i.e. Ujnj = ~ Ujnj = U.n j=1 etc. and the multiplication symbol * implies a convolution operation (e.g. a * b = b * a). Now the function Vs* is an undefined vector which we are at liberty to choose. We shall make this the time dependent fundamental solution of the linear adjoins Oseen operator subject to a unit impulsive loading in the jth direction acting at the position ~ = x, i.e. where the field point ~ coincides with the source point, x. That is Vs* sat~sfies the equat~ons E) Vs = Fs in Q Vs ~ 0 at infinity ~(14) together with the imposed initial condition vs * (x,0) = 0. Here the jth component of the external impulsive force Fs* is given by fsj = dsj ~ (~-x) d~t), such that Q VT* ~ . Vs dQ = C(~) vs(t,t) (15) where C(~) = 0, 0.5 or 1 depending on ~ ~ (Q U I), or ~ ~ Q respectively. Substituting equation (15) into equation (11) and using equations (8) and (10) we find that the integral equat~on becomes, C(~) vs(t,t) = ,[ { vs; * (Rj - vj U.n) - vj * Rsj } d + v * u - v v dQ - U v dQ 16 | sj ( j k j,k) | j si ~ ) Q Q assuming a zero body force i.e. f = 0. FUNDAMENTAL SOLUTIONS OR OSEENLETS TIME DEPENDENT SOLUTIONS Analogous to the steady state solutions (v = 0) derived previouslyE 12,13], the time dependent fundamental solutions or oseenlets satisfy the equations * * 2 * * Vsj - U.V Vsj = V Vsj + PS j + ~ j/` (t - x) ~t) v .. = 0 s~,~ Vs*j(t,O)=O. A solution to these equations can be derived using Laplace (L) and Fourier (F) transforms. That is, let ~y and ~ denote the Laplace and Fourier transform parameters respectively and an application of the sequence of operations LF~ ) to the previous equations gives ~ (17) JVsj + i\.U Vsj = - \2Vsj - i~jPS + {8Sj ei~ X/~2~)nl2} \. v.=0 ~ s~ where i = ~1~-l), n = 2 for a two-dimensional flow problem (s = 1, 2 = j) and n = 3, for the three-dimensional case (s = 1,2,3 = j). From these equations it follows that and 803 PS = - (il /~2) ei~ X/~2~)n/2 v* (6 (s\J) ei~ x/( 2~)nl2 [7 + il U + \2] By reversing the transform operations, we obtain[18] F(ps) = L { LF(pS) } = - i (\s/~2) e 6(t)/~2~) F(vsi)=(6sj- s 3)ei\.x e-(\ +il.U)1~2~)nl2

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or and Ps = n [,l >2 d) ].s ~ r = ~ - x (18) 00 ~Sj f - 1~ t +iA (r+ut)) v .=- I e dX (2~) 00 + ~ [T e-{\t+ix (r+Ut)} (211) 00 Three Dimensional Solution (n = 3) ,<2 - J,Sj (19) For a three dimensional fluid-structure interaction flow problem involving a field point ~ and a source point x, such that the distance r = l; - xI and rs (s = 1,2,3) denotes the sth component of r, the time dependent fundamental solution is Ps = 4 [1/r], s (20) V . = Sj e~ (r + U t) /4 t (4~t) I 1 [ 1-erfc ~ Ir+Utl/2 OCR for page 801
{l-e~( r)l2}{5s~ rsr~ (Us+rs/r) (Uj+rj/r) } 4~(U.r +r) r r~ U.r+r The vorticity distribution is given by C')sj= skj3 {2rk+r(Ukr+rk)}e~( r )/ 8~r where esk; is defined previously. MODIFIED INTEGRAL EQUATIONS Using the fundamental solutions derived in the previous section, the integral equation described in equation (16) can be written in alternative forms. For the submerged body, the surface boundary ); (=~,oo ~ ~ b) consists of the body boundary ~,b and an outer boundary ~0 at infinity. It can be shown that (i) on oo, Rj is constant, Vj = u; - Uj and When r ~ O. the velocity soluiion reduces to vj *l Rsi dE = - (uS-Us)/n, Vsj= ~ (6sj + 2 ~ corresponding to the equivalent Stokes flow solution and when r~oOwe have * ~ 0 (l/r2) outside the wake region, sj ~ 0~1/r) inside the wake region. Two Dimensional Solution (s = 1,2 = j = k) The oseenlet solution takes the form * 2 Ps = ~ rS / 2~r , * ~s~ - U.r/2 K ( /2) - [U.r e '2K0(r/2) + U.r ln r - r e / Kl(r/2) 2~ 2 rl2 (Uxr) I ~(~)e-(U.r)~4d~] sj o = Si e~Ur/2 Ko (r/2) + { Usrj + UjrS U ~Sj } { 1-0.5r e~ U r/2Kl(r/2) ~ (28) 2~r where Ko, K~ are modified Bessel functions of the zero and first ordertl83. When U = (1,0,0), this solution agrees with the form presented by Besshotl9] using an alternative approach. When r ~ 0, the previous solution reduces to v =- (- ~ . lnr+ s ~ ~or s~ 4~ s~ r and for r ~ oo, we have * r 0(1/r ~ outside the wake region, Vsj ' 0(l/r1/2) inside the wake region. In this case, the vorticity distribution is described by or the expression Cl) 3 = sk3 {Uk Ko (r/2) +-K~(r/2) } e~U r/2. r * J Vsj* (Rj-vjU.n)d~=O where n = 2,3 for two- and three-dimensional problems respectively. (ii) on 2,b, Vj= - U; and J vj * Rsj d): (27) = f 1 C(~ jus - J Uj*vsi dQ + J Uj*vsiU.n d~ and, Qb [b J u * v .dQ+lU. v .dQ ~s ~ s~ Qb - * = U.rO vs; nj d~ = (n-l) Us/n | vs; * Uj dQ = | vSj*u.r0 nj d~ - J vSj*u.r0 nj d~ Q ~o ~b = (n-l) Us/n - | vSj * u.rO n; d~ where r0 = x-y and y is any point in the domain and Qb denotes the internal volume of the body. The substitution of these expressions into equation (16) gives the integral equation relationships, C(~) VS (t,t) = US ~ C(~) Us ~ | vSj * (Rj+u r0nj ~d~ - | Vsj * (Vkvj k) dQ Q C(~) VS (t,t) = US - C(~) Us Jvsj * (Rj+ u.rOn j - Uj U.n~d~ +| Vsj k * (VkVj ~ dQ (29, Q C(~) VS (t,t) = US - C(t,) US-Ivsj * (Rj+u.rOn j - 0. 5U2nj~d~ * + vsj*(vx03)jdQ Q 805

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where ~ represents the vorticity of the flow. The basic unknowns in these formulations are the fluid velocity v and fluid force R which can be determined directly from the discretised forms of these equations. In fact, although the details of the terms in these equations differ from those occurring in the steady state problem, the general method of solution adopted in the latter can be suitably modified to solve the present integral equations. Further, since u, rO, U and n are all prescribed quantities we may treat Xj = Rj + u.rO nj - Uj U.n in equation (29) as the unknown and this integral equation reduces to C(~) VS (t,t) = US ~ C(~) Us-| Xj * vSj did job + | (VkVj) * VSj k dQ . (30) Q NUMERICAL SCHEME By discretising the continuous integral equation expressed in equation (29 or 30), the unknown X (or R) and v can be determined. Apart from the additional complication of the convolution time integral, the approach adopted here is similar to the one developed previously to evaluate solutions of the steady state flow problemEl2,131. That is, in the spatial idealization, the body boundary ~ b and fluid domain Q are discretised into mb elemental surface panels and ma elemental surface panels or volumes respectively and to assume that the unknowns satisfy prescribed distributions. Integrations are performed over each idealised element and the integral equation transformed into a set of simultaneous algebraic equations from which the unknowns on each panel and in each volume (or panel) are determined. Thus equation (30) can be written as mb C(E,) VS (`,t) = unit) - C(~) Us- 2, Xjm) * | Vsj did m=1 i) md + ~ (VkVj) * | VSj kdQ m=1 Q(,~ mb = unto - Ceil Us- ~ Xjm) * | Vsj dI m=1 an) md + ~ ~Vkvj, *f vsj nkdz m=1 2(m) (31) where );Q(m) denotes the boundary of the mth fluid domain panel. As can be seen, this expression involves an integration of the time dependent fundamental solution over defined surface boundaries and this can be represented analytically, as shown in the appendix. Following previously described proceduresEl2], the continuous integral equation represented by the discretised form of equation (31) can be expressed at each time step by general matrix equations written in the form - A X + B + C(V) = 0 - V - A'X + B'+ C'~V) = 0 } (32) where X and V are the unknowns to be solved. Although the individual descriptions of the matrices, A, A', B. B', C, C' are omitted, the matrices A, A' consist of linear elements, B contains linear and non-linear (i.e. products of terms) elements whilst B', C, C' represent totally non- linear expressions in the unknown V. In the steady state case, when the Reynolds number Re << 1, the non-linear contributions are negligible and can be ignored. Thus the unknowns X and V are obtained by direct solution from equation (32~. When Re ~ 1 and the non-linear contributions retained, a simple iterative scheme can be devised but when the Reynolds number is large, a more refined iterative scheme is required. This is based on the Levenberg-Marquardt algorithmEl2,20-22] which is a hybrid algorithm combining Newton's iteration method and the method of steepest descent. COMPUTATIONS STEADY STATE PREDICTIONS (V = 0) Since details of the steady state calculations are described elsewhereLl2,13], they are omitted here and the example included serves to illustrate the method. For a cylinder of diameter D' (=2A'), figure 1 illustrates the variation of the drag coefficient Cd (=Force/0. 5 p'U2D') with Reynolds number Re. Presented are predictions derived from an oseen flow-linearised model and a non-linear convective model as well as experimental resultsEl41. As can be seen, the linear results overestimate the other two data sets, with the non-linear results showing the better agreement with the experimental data. . NONL I NERR - - - - - - L I NERR X EXP. ~ TR I TTON ) 3 _ - ~- o LOG ( Re ) ~- Icy - _ _ Figure 1 A comparison between measuredly] and calculated (i.e. linear and non-linear flow models) steady state drag coefficient data Cd for a circular cylinder in uniform flows. In the linear model, the cylinder's surface was idealised by a distribution of 100 equally spaced viscous boundary elements though it was shown that a 10 element distribution produced similar convergent solutions and both sets of numerical results compared very well with Oseen's theoretical prediction as given by LambE23, Re 13. Further, in the limit Re ~ on, Cd ~ 2.28 again indicating an overestimation of the experimental data. 806

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In the non-linear model, 100 equally spaced elements were distributed over the cylinder's surface as well as a similar number over an imposed surrounding outer boundary six diameters (R=6D) away from the surface. The enclosed fluid domain was idealised by 400 panels with the distribution showing a greater density near the cylinder and decreasing radially. Figure 2 shows the computed flow field around the cylinder (Re = 200) with the vortex wake pattern clearly defined. .. ==~ ~ ~ ~- Figure 2 Computed Oseen flow field around a circular cylinder, Re=200. This investigation showed that calculations of the fluid actions, pressure distributions over the cylinder's surface etc. are satisfactorily determined by the non-linear convective model and the global solutions are relatively insensitive to the mathematical model and idealization. However, calculations of the flow velocity field are far more sensitive to idealization and the truncation distance R of the surrounding outer fluid domain boundary. The latter must be placed at a sufficient distance from the cylinder to allow for the complete generation of the vortex wake pattern. Therefore, if information on fluid actions is required only, a much reduced computational model can be adopted, significantly decreasing the computational effort with a relatively small numerical error introduced if the truncation distance R is taken to be some value R > 2D (say). NONSTEADY PREDICTIONS (V = 0) Oseen Flow-Linearised Model A linearised mathematical model can be developed if the time dependent fundamental solutions are distributed only over the cylinder's surface. The convolution time integration is retained in the modified integral equation of equation (29) but the troublesome non-linear convective contribution is discarded. For the time history predictions presented in figure 3, the cylinder's surface was idealised by a distribution of 40 equally spaced viscous boundary elements each containing at its centre a time dependent oseenlet. The co .mponent velocities us, u2 defined in the body frame of reference are shown in figure 3(a). That is, the motion in the longitudinal direction of the body (i.e. surge, ups shows the body accelerating to a fixed speed, remaining steady until it experiences a slight blip before returning to the previous steady condition. At the same time, in the transverse direction the sway component u2 is sinusoidal. This simple example serves to demonstrate the arbitrarily selected motions (e.g. a prescribed manoeuvre) which can be introduced into the mathematical model without difficulty, though it bears a similarity to a PMM test procedure. The time histories of the drag coefficient CdX (=R~) shown in figures 3(b-d) follow the variations of the motion units in each of the Reynolds number flows Re = 2, 40, 200. At each transition the coefficient exhibits an over- or undershoot tendency whereas the transverse force coefficient Cdy (=R2) retains the oscillatory behaviour of the prescribed motion input but with a phase shift depending on Reynolds number. in ~6 2 8 L) ~ \ ~ 4 I ,' , \ I,, \ \ , \ , ' , so " coo (C) In ~an \ / Cdu1 (d) V / \ / \ to \ / \ ,, ~\ \ / \ \ ~ \ 720 1! \ to \ / \ / \ / \ , Figure 3 The time histories of the prescribed manoeuvre of the circular cylinder and the associated calculated Oseen drag coefficient CdX and sway transverse force coefficient Cdy for different Reynolds number flows. The abscissa denotes the number of time intervals passed into the calculation. (a) the surge units and sway u2(t) motions, (b) Re=2, (c) Re=40, (d) Re=200. 807

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Figure 4 illustrates a series of snapshots of the flow field behind the cylinder observed from a fixed reference position. This sequence of frames at 10, 20, 3(),..., 120 times the time interval increment clearly shows the vortex wake forming and decaying behind the cylinder as it undergoes the prescribed manoeuvre shown in figure 3a. The continuous line behind the cylinder indicates the path of the cylinder, in a flow field associated with a Reynolds number Re = 40. 10 20 to 30 40 50 60 70 80 Q- o i o o a To - 90 100 1 10 120 of C: - Figure 4 A sequence of snapshot frames at 10, 20, .... 120 times the time interval increment illustrating the formation of the vortex wake behind the cylinder associated with the calculation Re=40 in figure 3. In both the steady and unsteady oseen flow calculations, no difficulties were encountered in the numerical scheme of study for any chosen value of Reynolds number (e.g. tom, etc). However, it must be emphasised that from the evidence available the results derived from this linearised model are expected to overestimate the experimental data and, as we shall further show, the results obtained from the non-linear convective model. Non-linear Convective Model In the results presented in figure 5, the non-linear convective term in the modified integral equation, i.e. equations (29,30), is included in the numerical scheme of study. To do so requires distributing viscous panel elements into the fluid domain and this greatly increases the computational effort needed to provide solutions. In fact, this necessitates the full solution of the non-linear coupled matrix equations in equation (32) whereas the oseen flow solutions are obtained from a much simpler linear matrix modelEl2,131. The cylinder's surface was again idealised by a distribution of 40 equally spaced viscous boundary elements, each containing a time dependent oseenlet and in a similar procedure to the equivalent steady state calculations, 400 panels were distributed into the fluid domain contained within an imposed surrounding outer boundary at R = 6D (say). The prescribed manoeuvre displayed in figure 3(a) was again chosen and figures S(a-c) show the calculated force coefficient components as a function of the number of time step intervals and Reynolds number Re = 2, 40, 100. It is seen that these time histories show similar trends to the equivalent linear predictions in figures 3(b-d) following the variations of the 808

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input parasitic motions but, in comparison with the linear findings, their values are reduced. Note, however, the ~ " oscillatory variation now evident in the CdX time record , , ~ ~ ~ ~ ' created by the oscillatory sway motion. This again ~, ' , _ - _ ~ ~ illustrates the influence of the non-linear convective term in ~t , ~ _ _ _ ' ~ ~l l the mathematicalmodel. ~ ~ ' __ ' ~ I, ' Figure 6 shows a limited sequence of pictures " "t If',,_ 'it i, , illustrating the creation and decay of the vortex wake flow ~ ~" ~, ~ ~ '~ ~ ,',, ~ I, i, ', ~ field around the cylinder as seen by a fixed observer.' ~ ~ ~ ~'~t ,' - ~ ~ _ 16 -- - ' ~ ~ - ,, ~, , ~ \ , ~ _ - / , ' 10 \\ // 40 \ ,/ 80 \ ,/1 20 'A ~ ,' " , ~-` ~ ~ ~ ~ ~t ~ , 40 \ / 80 \ ~120 ~ ~ ~ ~ _ ' / I 20 3 _ _ Cdx I (C) 3, _ ' ~ i'" '. ' ~, ,,~tt'x'~-~ hi,, " Figure 5 For the manoeuvre illustrated in figure 3(a), ~ ~ ,, " `-~ ~ , ' , this figure shows the associated drag ' ~ ~ "a-- ,, coefficient CdX and sway transverse force ' ~ ~" ~ _ - , coefficient Cdy for different Reynolds number ~ ~ _ ~ flows calculated from the non-linear convective ' \ - model. The abscissa denotes the number of ' time interval increments passed into the simulation. 30 (a) Re=2, (b) Re=40, (c) Re=100. 809

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~ ~ - ~ - t~ Air it's Ott / l l ~- ~" ~- - ~- - ' ~- ~- 40 70 t _ 1 _ t ~_ ' t ~/ ~_ _ ~ i~ ' I it ~ ,'- ~ ~ _ - - ; ~1. = _ _ ~_ _ ~_ ~_ ~_ 50 80 ~t _ ~_ t t _ t ~ , t t _ _ ~ t ~ ,, _ t ~ ~ ', - ~t , t ~ - - __- As, A= --' -','~gT~J , - ,'. Ads ,' , , ~ , ~ ~ ~ ,~ __ _ _ , , ~_ - ~, , , , , ~_ I ~_ , , , ~_ _ _ ~, ~_ ' ~_ , 60 90 810

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, ' ' ' - _ Figure 6 \ t ' ' , ' t , , , ,, _ _ '` ' "`'!,',',,''," 2 "i J ~ J J I t 1 l ~ ~ ~ ~ ~- , _ 1 1 ~ ~ ~ ~ ~_ _ _ _ - - 100 t t t t t t ~ J - ~ l ' ~ \~-\ ~- ~t J l J l 1 J J ~ ~ ~ ~- 1 1 ~ ~ _ _ , _ _ , 1 ~_ _ _ 110 t t l ~J ~ l , ~ ~ ~ \~N _ ~ ~ ~ ~ _ l ~ ~ ~ ~ ' ~_ 120 , ' ~ - 811 A sequence of snapshot frames at 10, 20, .... 120 times the time interval increment showing the generation and decay of the vortex wake behind the circular cylinder associated with the calculation Re=100 in figure 5(c) and also figure 3(a). CONCLUSIONS The hybrid analytical and numerical viscous boundary element approach using time dependent oseenlets described herein, allows the predictions of the fluid actions and flow fields associated with arbitrary shaped bodies moving in a prescribed manoeuvre in an incompressible, viscous fluid. Although the method is demonstrated using a two-dimensional simple shaped body, the concepts introduced remain valid when tackling three-dimensional fluid-structure problems. However, in the latter, the presentation of information - especially a description of the flow field - becomes more difficult and the computational effort greatly increases. The Oseen flow-linearised model produces over- estimates of the fluid actions and flow fields but it is easy to apply and, since it provides a 'broad brush' picture of the fluid-structure interactions, in engineering terms, it produces a reasonable first insight and solution to the problem. The non-linear convective model is computationally more time consuming though the evaluation of the fluid actions is obtained from a relatively robust numerical scheme of study. However, because of the sensitivity of the flow field calculation to panel idealization, truncation distance etc., the preliminary calculations presented serve to illustrate the applicability of the viscous boundary element approach to evaluate the time dependent fluid actions and flow fields associated with bodies manoeuvring in an incompressible, viscous fluid. ACKNOWLEDGEMENT We gratefully acknowledge the support of the Science Engineering Research Council, the Ministry of Defence Procurement Executive and the encouragement of the staff at Admiralty Research Establishment (Haslar). We are indebted to Mrs Christa Steele for her typing (and retyping) of this manuscript. REFERENCES 1. Duncan, W.J., The Principles of the Control and Stability of Aircraft, Cambridge University Press, Cambridge, 1952. 2. Etkin B. The Dynamics of Flight Wiley New , . . . York, 1959. 3. Mandel, P., "Ship Maneuvering and Control," Principles of Naval Architecture, (ed.J.P.Comstock), Society of Naval Architects and Marine Engineers, New York, 1967, pp. 463-606. 4. Bishop, R.E.D., Burcher, R.K. and Price, W.G., ''The Uses of Functional Analysis in Ship Dynamics,' Proceedings of The Royal Society London, Vol. A332, 1973, pp. 23-35. 5. Bishop, R.E.D., Burcher, R.K. and Price; W.G., reapplication of Functional Analysis to Oscillatory Ship Model Testing,ll Proceedings of The Royal Society London, Vol. A332, 1973, pp. 37-49.

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6. Booth, T.B. and Bishop, R.E.D., "The Planar APP Motion Mechanism," Admiralty Experiment Works, ENDS Haslar, 1973. 7. Burcher, R.K., "Developments in Ship Manoeuvrability," Transactions Royal Institution of Naval Architects, Vol. 114, 1972, pp. 1-32. 8. Clarke, D., "A Two-Dimensional Strip Method for Surface Ship Hull Derivatives: Comparison of Theory with Experiments on a Segmented Tanker Model," Journal Mechanical Engineering Science, Vol. 14, 1972, pp. 53-61. 9. Mikelis, N.E. and Price, W.G., "Calculation of Hydrodynamic Coefficients for a Body Manoeuvring in Restricted Waters Using a Three-Dimensional Method," Transactions Royal Institution of Naval Architects, Vol. 123, 1981, pp. 209-216. 10. Mikelis, N.E. and Price, W.G., "Calculations of Acceleration Coefficients and Correction Factors Associated with Ship Manoeuvring in Restricted Water: Comparison between Theory and Experiments," Transactions Royal Institution of Naval Architects, Vol. 123, 1981, pp. 217-232. 11. Price, W.G. and Tan, M., "A Preliminary Investigation into the Forces Acting on Submerged Body Appendages," Proceedings of the Conference on Ship Manoeuvrability. Prediction and Achievement, The Royal Institution of Naval Architects, 1987, paper 13. 12. Price, W.G. and Tan, M., "The Evaluation of Steady State Flow Parameters Around Arbitrarily Shaped Bodies Using Viscous Boundary Elements," Report 1/89, 1989, Department of Mechanical Engineering, Brunel University. 13. Price, W.G. and Tan, M., "The Evaluation of Steady Fluid Forces on Single and Multiple Bodies in Low Speed Flows Using Viscous Boundary Elements," International Union of Theoretical and Applied Mechanics Symposium on The Dynamics of Marine Vehicles and Structures in Waves, June 1990, Brunei University, (Also Elsevier Press, 1991~. 14. Tritton, D.J., "Experiments on the Flow Past a Circular Cylinder at Low Reynolds Number," Journal Fluid Mechanics, Vol. 6, 1960, pp. 547-567. 15. Thom, A., "The Flow Past Circular Cylinders at Low Speeds," Proceedings of The Roval Society London, Vol. A141, 1933, pp.651-669. 16. Fornberg, B., "A Numerical Study of Steady Viscous Flow Past a Circular Cylinder," Journal Fluid Mechanics, Vol. 98, 1980, pp. 819-855. 17. Fornberg, B., "Steady Viscous Plow Past a Circular Cylinder up to Reynolds Number 600," Journal Computational Physics, Vol. 61, 1985, pp.297-320. 18. Abramowitz, M. and Stegun, I.A., ea., Handbook of Mathematical Functions, Dover, New York, 1972. 19. Bessho, M., "Study of Viscous Flow by Oseen's Scheme, (Two Dimensional Steady Flow)," Journal Society of Naval Architects Japan, Vol. 156, 1984, 20. Levenberg, M., "A Method for the Solution of Certain Non-linear Problems in Least Squares," Ouarterlv Journal of Applied Mathematics, Vol. 2, 1944, pp. 164- 168. 21. Marquardt, D.W., "An Algorithm for Least Squares Estimation of Non-linear Parameters," Journal of Industrial Applied Mathematics, Vol. 11, 1963, pp. 22. Twizell, E.H., "Numerical Methods. with Applications in the Biomedical Sciences," Ellis Horwood and John Wiley, Chichester, 1988. 23. Lamb, H., "Hydrodynamics," (6th ed.), Cambridge University Press, Cambridge, 1932. 812 Before writing equation (31) in matrix form, we need to Integrate the time dependent fundamental solution over an elemental surface panel or volume. For the two dimensional problem under discussion this may be achieved as follows. Uxt ~,~s Figure 7 Schematic illustration of the panel, field point, etc. and symbol definitions. Figure 7 illustrates a panel with end points (1,2) lying in the direction of a and ~ is a field point connected to the source point x on the panel by the vector r = x - (. The vectors dl, d2 are as shown and the unit normal n points out of the panel. If we let r=r+Ut, dl=dl - Ut, d2=d2-Ut where and Because r= 2 (BaO-n), 86 [- OCR for page 801
r 1 ~^ ~ nnN ~ a n 1 na ~ _ = fi_nn = r2 O O _ O O _ O O r J ~1 -~52 - r 2/4t e d~ t = 0 S(~/t)ll2 e- (n dl) /4t [erf (aO d 1/2tl/2) - erf(a~' . d212t 12)] then the surface integral involving the two-dimensional, time dependent oseenlet is given by the analytical expression 2 2~ | vSj d~ = | { t (6 --) e~ r /4t ~1 +2( 6+2rr)(1-e~r /4t)}dD r2 r4 = ~-(nn-a~aA) + 26 (a n+na^)1 (1-e r2/4t) | <52 r2 --U U' r2~ V u~-~ -~71 ,[ t {~~ - 2 +(aOaO-nn) 2 ~ (aOn+naO)4E~ }dJ3 ~1 = 0.5(~/t)l/2 e~ (n dl) /4t{erf (aO. dl/2tl/2 - erf (aO.d2/2t /2) } aOaO aO. d2 dO2/4t + (aOaO-nn){ 2 (1-e d2 ~. +(aOn+na`}) {-(l_e-d2/4ti d2 -2 (l-e-d1 /4t)} )- 2 1 (l-e~d1 / )) dl Although this is an unwieldly analytical expression, it avoids the numerical integration of the oseenlet over the surface panel but, unfortunately, its form does not readily permit the convolution integration to be reduced to analytical expressions. 813 l

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DISCUSSION Gerard Fridsma General Dynamics, USA The statement was made about the importance of Reynolds number on model testing of submerged bodies. While indeed answers are being obtained on the testing tunnel and towing tank, these are of a qualitative nature to understand the nature of the flow and production of loads. For small cross-flow angles (yaw), the loads are linear and not Reynolds number dependent in this range. As one goes to higher cross-flows (yaw angles from 10 to 20), vortices are generated from the hull which create serious nonlinearities in the trends of the loads. These must be dealt with for a full maneuvering submerged body which means Reynolds number must be dealt with, since separation and vortex generation is very much dependent on Re. W\\~\~: ~ ~ ' 4.5 4ao AUTHORS' REPLY We thank Mr. Fridsma for his contribution to this paper and we agree with his overall observations. Our comments relating to a submarine hull were based on steady state calculations performed on a two-dimensional cross-section with sail plane or fin. This was simply idealized by a circular cylinder and appendage as illustrated in the following figures. These also display the steady flow field around the section and show the variation of the steady drag and lift coefficients with Reynolds number. In the region lee < 103 (say) the curves tend to flatten indicating that large changes in the Reynolds number produce relatively small changes in these steady forces. These preliminary findings may provide a simple explanation why model scale and full scale submarine experiments produce a measure of correlation even though true scaling is impossible to apply in practice. Although a free running or towed model is geometrically scaled correctly, it must be of a size to contain the instrumentation measurement packages and not too large for the towing tank or maneuvering basin. Thus constant Reynolds number and viscosity coefficient imply that the model's forward speed must be set at Um = (LFS/L,,,) UFS, where subscript FS denotes full scale value. Practica fly, in a towing tank, this relationship is impossible to fulfill and so a model forward speed is chosen as high as safe powering allows within the confines of the test facility. Thus the Reynolds number for the model and full scale differ, but if both experiments are performed at Reynolds numbers lying within the flat portion of the drag and lift curves, then it can be expected that steady state predictions for model and full scale would display reasonable correlation. In fact, from the values of the steady state forces at the appropriate Reynolds numbers for a model and full scale submarine a simple correction factor could be deduced and incorporated into maneuvering prediction mathematical models. The calculated fluid flow field around a body appendage configuration Elm). - - - DRRG ~ ~ F T 3.5 v) an ~ 3.0 - IL ~ 2.S to LJ ~ 2.0 LL .5 1.0 0.5 / \ / \ - . . . 2 3 4 REYNOLDS NU1118EQS ( LOG(Re) ) s 6 814 The calculated variation in the lift and drag coefficients with Reynolds number.