DTNSRDC-82/077 ONAL PRINCIPLE ASSOCIATED WITH A LOCALIZED NUMERICAL SOLUTION ADY FREE-SURFACE FLOWS DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20084 A VARIATIONAL PRINCIPLE ASSOCIATED WITH A LOCALIZED NUMERICAL SOLUTION OF UN- STEADY FREE-SURFACE FLOWS DOCUMENT COLLECTION APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT September 1982 DTNSRDG-82/077 MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER 00 TECHNICAL DI athe hy OFFICER-IN-CHARGE CARDEROCK OFFICER-IN-CHARGE ANNAPOLIS SYSTEMS DEVELOPMENT DEPARTMENT AVIATION AND SURFACE EFFECTS DEPARTMENT SHIP PERFORMANCE DEPARTMENT 15 COMPUTATION, MATHEMATICS AND LOGISTICS SERA es STRUCTURES DEPARTMENT PROPULSION AND | SE Oeics AUXILIARY SYSTEMS eal DEPARTMENT 19 SHIP MATERIALS CENTRAL ENGINEERING INSTRUMENTATION DEPARTMENT on DEPARTMENT MBL/WHOI AAA 0 0301 0037131 4 NRC NDW-DTNSRDC 5602/21 (2-80) Dept. 9 UNCLASSIFIED ie OCT 1 2 1982 eee SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE T. REPORT NUMBER 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER DTNSRDC-82/077 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED A VARIATIONAL PRINCIPLE ASSOCIATED WITH A ae LOCALIZED NUMERICAL SOLUTION OF UNSTEADY are, » AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(S) Bev Nam . PERFORMING ORGANIZATION NAME AND ADDRESS 10. Aaah ai a Monee: TASK * . UNIT NUM David W. Taylor Naval Ship Research Program Element 61153N and Development Center Task Area RRO140302 Bethesda, Maryland 20084 Work Unit 1542-018 - CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE September 1982 Office of Naval Research (ONR-432) is NUMBERID EE ACES » MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) | UNCLASSIFIED 1Sa. DECLASSIFICATION/ DOWNGRADING SCHEDUL - DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED - DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES . KEY WORDS (Continue on reverse side if necessary and identify by block number) Variational Principle Numerical Ship Hydrodynamics Finite Element Technique Nonlinear Theory Free Surface Convolution Functional . ABSTRACT (Continue on reverse side if necessary and identify by block number) In this report, a variational principle for unsteady body wave problems is treated both with and without a convolution integral and with both linear and nonlinear free-surface conditions. Functionals are obtained for the numerical computation of unsteady flow fields near a body that moves on or beneath the free surface. This formulation can be applied to ship hydro- dynamic performance problems of water entry and body slamming, as well as to arbitrary body motion. DD . oon, 1473 EDITION OF 1 Nov 651s OBSOLETE UNCLASSIFIED S/N 0102-LF-014-6601 SECURITY CLASSIFICATION OF THIS PAGE (When Data Bntorad) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) TABLE OF CONTENTS Page INOPASMIONiswiter icy) aay (euenct= rel feted “\ shee) or cl fst Poll e) Wiomioe, Neleiest lewetin dire toeoree eet Mee UAeRME ols iv JNOSHMRINGIN GS a aa Ray hg ceo colsch oo) clon Geno data, cod ay\On earn vo-fos ce ono) Cola) vofoh ole (a a ADMINISTRATIVE, TENE ORMATION sa tse) uses ce veuuiog! cievte Vs pteq vu bomreue) Youlow cl ie imey lion 1) snare ie 1 JONAUSOVDIOLOIMILONS Goh Geo G6 6 ib doo 6 6 Gb oo6 we soo 6 Cele G6 bose 6 or ile INCOM IGIUNI NNR TINOITHIONE 5 5 GG Go 6) 6 6 6 8 Od Goo 6 Go 6 odd of a 8 4 of Gee 2 IDJUMIMAR INOIIH BGS 6 6 fo o 6 6 865.0 6 5 6 Ob Geo ed 6 SB so 8 ot 5 io 6% D (WE, LORS THOIMNVOIHOMEIOIN GS. G Gg 6 AG 6 dee ol 6 ogo Oe G6 be Boe o 6 Goo oom oe 7 SUNLUSTOMED YW WOMEION iors we A Goo) wenneo co fob Cleon em o do o oO © G@ oO obo oo oc 10 ICI IN Gal SoG ol od ob GG oo Me 6 OG Go eo 6 ob ced 6 o 6 HGS Go 10 CONCLUDING: REMARKS! 52 ar a Gn “ei seis © ie bo ee ie ie eels Ge) se Se es Ge als 2 TOD OGIO TR oe ene heen 8s ce ooo 0 Oo §. ovo Ge oc coe 8) pee ooo ea iehion loo BA UrG OTS ic 113} ae iat NOTATION D Fluid domain f; Function defined in Equation (23) g Acceleration of gravity By Function defined in Equation (24) h Body or free surface defined in Equation @)) J Functional m Source distribution Pp Pressure distribution Sp Free surface Sy Interface of near and far fields S. Body surface t Time Viet Rectangular Cartesian coordinates 0) Variation 0 Water density T Time a) Potential w Function defined in Equation (21) w Frequency defined in Equation (21) Subscript i The ith order n Normal derivative toward fluid fe) Projection on the z = 0 surface t The time derivative iv The z derivative Near field, the first order Far field, the second order ABSTRACT In this report, a variational principle for unsteady body wave problems is treated both with and without a convolution integral, and with both linear and nonlinear free surface con- ditions. Functionals are obtained for the numerical computation of unsteady flow fields near a body that moves on or beneath the free surface. This formulation can be applied to ship hydro- dynamic performance problems of water entry and body slamming, as well as to arbitrary body motion. ADMINISTRATIVE INFORMATION The work reported herein has been supported by the Numerical Naval Hydrodynamics Program at the David W. Taylor Naval Ship Research and Development Center. This program is jointly sponsored by the Office of Naval Research and DINSRDC under Task Area RRO140302, Work Unit 1542-018. INTRODUCTION In the early 1970's the David W. Taylor Naval Ship Research and Development Center (DTNSRDC) recognized the demand for advanced numerical methods to predict the hydrodynamic performance characteristics of naval ships, particularly when classical methods proved inadequate. Thus, in 1974 the Numerical Naval Ship Hydrodynamics Program was begun at DTNSRDC. Under this program the author previously investigated the steady ship-—wave problem using a variational principle associated with a localized finite-element Pochndaeee This method is useful particularly to analyze the flow field near the ship in detail; in the far field, the Michell approximation can be used. This report extends the problem to the unsteady case. For both the steady and unsteady problems, the simple calculation is for a linear free surface condition with exact body boundary conditions. An iterative method is needed for a nonlinear free surface condition. However, in the unsteady problem, the variational principle requires an integration with respect to time using the initial conditions; in this instance, a convolution integral is useful. The variational principle for the unsteady body wave problem with exact body boundary conditions is treated both with and without convolution, and with both linear and *A complete listing of references is given on page 13. ai nonlinear free surface conditions. For the linear free surface problem, a func- tional for the variational principle is obtained with a convolution rather than a general integral. The convolution cannot be applied to a nonlinear free surface problem; the condition is required for large values of time. A nonlinear solution derived using an iteration scheme having the linear convolution form is also dis- cussed. The time integration can be eliminated if the motion is sinusoidal. This formulation can be applied to problems of water entry and body slamming, as well as to arbitrary body motion. NONLINEAR PROBLEM Since problems dealt with here can be generalized easily to three-dimensions, for simplicity we first consider a two-dimensional problem in the rectangular Car- tesian (x,z) coordinate* plane. When a body whose surface is represented by S. [z=h(x,t)] | @) enters the water surface Sa (Ze= 10), bes a = h(x,t), t > 0) at time t = O, or when a semisubmerged or fully submerged body starts to move at time t = O and either exits the water or stops moving at t = th» then the boundary conditions for a velocity potential $¢ are as follows: ¢ = >, = 0 for t < 0 everywhere + (1b)? - 6, + gh=0 oS (2) h - VoV (h-z) = 0 h_ - ® eo = 0 t n x Here, S(t) is the submerged body surface varying with time t, and n is the normal direction into the fluid. We consider potentials 1 in the domain Dy and >, in the *Definition of notations are given on page iv. domain Dos where dD, is the near field including 85> and Dy is outside of D,- Then at the interface Sy of D, and Das we need to have (4) ¢,--¢ n 2n The outer potential 5 in Ds is assumed to satisfy the linear free surface condition ate t SPon © o For such 5 we know the time-dependent Green fundetor Now we will construct a Lagrangian for the previously described problem, con- sidering the Lagrangian that feateee used ic ‘ 2 1 re x i -| \j (5 Ma % 1-41, ) azexet +f | g > dxdt a | (0,- 5 by) bo. dzdt (6) a 0 Dd, 0 Sip 0 Sy where P> %, and h vary with time, and t is a sufficiently large time after the body has either exited from or come to rest in the water so that we can safely assume that the variation bo), Sb, aue 5b, vanishes at t = tT. It will be shown later that the use of a convolution integral necessitates only the initial condition without the Condition at t = T. Since hiGcat ye ~niGc, t) ma bdz = ods + $(z=h) a (7) we have re h(x,t) h(x,t) { fx] >. dzdt I aE al odz — o(z=h) hy dxdt 0 heat) T AE “| ee | | dt (z=h) he dx 0 or Ag cr 5 {| ), dzdxdt = S| | h, ¢dxdt +f bdxdz (8) Dis Ce) Q ss. D, (t=t) Now we take a variation of J in Equation (6), and use the Green theorem, and the He eye aL condition at large t = T obtaining T 1 6J -{ | € Vo, Vy ,t2h ) Oh dxdt 0 Sie au 40 “| il vo, 89, dzdxdt I { (19/65 *,) 6h, dxdt 0 0 3, ,Us. Di 1F i ic =| | (>, +45,) 66, dade = | (¢,-$5) 665, dadt 0 Ss; 0 Sy i 1 + “| J (50. P5409 56, ) dzdt = 0 (9) 0 Ss However trom the identity, ff (9° 865-50" (66,)} ards -| {$, 66,-$5(8b,,)} ds = 0 S 7) Sor and from the linear free surface condition on Sop Equation (5), and the condition at t = T, we have € i | J (9 55,,-$5565,) dadt = =| A (8$505,-¢5505,,) dxdt 0 Sj 0 Se (10) iG . Al SUT he STP ores a | (8$.565,7$95¢5,) = dx = 0 ope w Sop so that the last integral of Equation (9) vanishes. Since 5by5 5b, Sbo0> and 6h are arbitrary, we obtain from Equations (9) and (10) the corresponding time- dependent, free surface boundary value problem represented by Equations (2)-(4) together with the Laplace equation for ¢. Therefore, solving the variational problem with the functional J of Equation (6) is equivalent to solving the Laplace equation with the boundary conditions as set forth in Equations (2)-(4), provided we assume that 6b, = 5b, = Ob, = 0 at t=T LINEAR PROBLEM If we assume a linear free surface condition in both Sip and Sop and keep the exact boundary condition [Equation (3)] on the body surface, Paper ie! nH ad then we may use 18 a {fa Vo,V$, dzdxdt - ak [ $191 ep dxdt OD iL {Jom f] where S is the projection of S, on z= 0. When we take the variation of J, we have Fo S13 AD. Sace Veoh edule dsdese Oe ee deat me Tele eee pa | UA aes, lee) oe anny SN 1 (i> 2&2) $,,dsdt (12) ie ( o,, ve -,) dxdt T c -| j ($4 -%5) bo) dzdt -| j (6) 5,257 dzdt (12a) 0 Sy 0 Sy Here, in addition to Equation (10), the following equation holds J js Care dxdt -f J Tee dxdt f i ee dxdt als Z 2) a +f 9750 OO ter OX 2 0 “Ss 0 Fo ene ae (13) Thus, as in the previous section, we can easily derive the corresponding linear free surface boundary value problem using Equation (5) with the exact body boundary con- dition in Equation (11). USE OF CONVOLUTION Equations (6) and (12) are Lagrangians in a time-dependent, two-dimensional space with nonlinear and linear free surface conditions, respectively. They could be localized in space but not in time. Namely, we had to specify the conditions on Se at t = 0 and t = Tt with a sufficiently large T. On the other hand, we did not require the initial condition un = 0 at t = 0. In addition, such time T when 1 = 0 on Sp may be too fleeces for practical use. For linear free surface boundary condi- tions, we can treat our variational problem in the same way as those who have treated variation principles for linear initial value problems using convolutions defined by i * = oe oT 5 :{ >, (x,z,t) (85257 t) dt 0 (14) ab ad ao ao = 1 2 1 D 15 * VO5 ~ Ox z ox us Oz ; Oz We change Equation (12) to Ta ye. eo deol] Caren ecee 2 al 1 22 il det an SG 5 ak ; | he x %) dx | (0° 7 +, * Pon dz (GLE) ne Si If we use the identity relation and the initial condition %) =o = 0 at t = 0, instead of the conditions Ot = 0 on lt Sp at t = 0 and t = Tt with large T used in the previous section, we obtain, for any time, T T 1G s| 1 * ee dx -| 5o, (x, t) Prep Hott) dxdt | | >, (x,t) 5b pp He T-t)dxdt 0s 0s Fo Fo Fo T = 2 ( 6, (x,t) dy pp So Tot dxdt 05 Fo +f {($, (x,t) S$, , (x, t-1)-$,, (x, t) 84, (x, t-t)} 14 dx S Fo il ho o> T | dp, (x,t) by pp Ho Tt) dxdt S Fo (16) HI N n> OQ a ke > rR ct Gr a. * where T need not be large, and x represents a point on the free surface z = 0. Thus, we obtain as in Equation (12a) 2 x : D i Sao f2. - | 6b, * (1, hy+1 - hy] dx S Ss || (5-95) * 86, dz - i (b, 45.) * 60,42 (17) eT aT To obtain Equation (17), we used a convolution expression of Equation (16), where we can use the initial condition 4 = =1 0; Ve Since Soy > Soo, and Soo, are arbitrary, we obtain the corresponding equations for a time-dependent linear free surface boundary value problem that has a unique solution. If we lift out Sy so that D, occupies the entire fluid domain, then the last integral of Equation (15) ieee The resulting equation appears much simpler than that obtained by Mapeae! due to a simple difference in the treatment of the free surface condition. Equation (15) does not give the wave height as a natural boundary condition, whereas Murray's corresponding equation does. However, from h = $,/8 the wave height can be obtained. If we consider eigen solutions that satisfy only the Laplace equation; the linear free surface condition, Equation (5); and the radiation condition in dD, U Dy» then >, which satisfies the body boundary condition in Equation (11), can be derived from Equations (15) and (17) by using 1 Vid. = —_— — * J | (3 Oe hyt1 ny] odx (18) s s When we know a functional whose minimum value is attained by the solution, we can find the solution numerically by such methods as the finite-element eechatiques F P 9 or singularity method. For example b= D7 a9) (19) where o, is the Green function, which is available for this problem for a source distribution on the body surface. The source distribution m; will be obtained from soltuion of the simultaneous equations, SO . fa1'2,°.2N (20) SINUSOIDAL MOTION If we consider a sinusoidal ship oscillation such as h =f Bawe for S, of Equation (1), we substitute i iwt into Equation (12), integrate with respect to t, and obtain -iwt Al, Ne 2. e J= Jy = > Vb, dzdx - oe vy dx Si Sp +) fb, dx-] (vw -tu,)u, 4 (22) i ices ( P22 5) Dass © ji This is exactly the same Lagrangian that Bai and Yeung. used. Working from Equation (18), we can apply eigen solutions to the whole field by using i =| (5 ¥,-F) ds where h,/ vno+1 ape A similar functional was used by Sao et Fale? to solve the problem of a heaving oscillation of a dock. ITERATIVE SCHEME For problems with the linear free surface condition, we can completely localize the numerical scheme in DS with 0 < t < Tt for any T with the help of the convolution form. However, for nonlinear problems, the finite-element technique has to rely on : P “ : saa oe an iteration. We may thus use an iterative free surface condition on z = 0 10 OVS gel f=: 0 (23) gh, - bat +g, = 0 (24) where, for the first order perturbation solution, fy = 0, 8) = QO and, for the nth order, ae and g, are known functions of 5 of the (n-1)th or the lower order solutions. Then the Lagrangian for each f. is 1 i = —- * —_—— oe {J Paes eae | oie uae & Dy (z=0)ND, ae * * | $,, *f, det fo, 4, ax (z=0)9D, oa oJ, = 0 (25) where the solutions for 1= 1, 2,...n - 1 should be used to determine the solution when i = n. Equation (24) gives the wave height h for each i. If we specify a time-dependent, free surface pressure distribution p on the projection oan of S, to z = O instead of hy in S. we may use £, = 8) — p/P OD ee in Equations (23) and (24) for the first order, where p is the water density. lel! Although we have discussed the two-dimensional time-dependent problem, Equations (19) through (24) can be extended to the three-dimensional time-dependent problem. | al il = = *k Sek D That is l (2=0)ND, - \| , * fs dxdy (fr, * 4 dxdy (z=0)ND, alt -{{ (¢)- 2 b,)* 6, de =F and soon. CONCLUDING REMARKS We have formed functionals with both linear and nonlinear free surface boundary conditions. For the former but not the latter case, we could apply a convolution integral. However, the body boundary condition is satisfied exactly in both cases. In many cases, the flow field near an arbitrary body is of interest, and eigen solu- tions with linear free surface conditions are known. Even in any large unsteady motion such as ship slamming, the free surface condition for a short period in the beginning may be linear, then the convolution may be applied in the early time period. Especially in the slamming problem, the peak pressure is known to be reached early in the beginning and estimation of the early pressure distribution on the slamming body is required. With this functional, we can find the solution for an arbitrary body numerically by such methods as the finite element technique or singu- larity method. Thus, a wide application of such functionals can be expected. 12 REFERENCES 1. Yim, B., "A Variational Principle Associated with a Localized Finite- Element Technique for Steady Ship-Wave and Cavity Problems," Proceedings of the First International Conference on Numerical Ship Hydrodynamics, David Taylor Naval Ship R&D Center (1976). 2. Wehausen, J.V. and E.V. Laitone, "Surface Waves," Encyclopedia of Physics, Springer-Verlag, Berlin, Vol. IX, pp. 446-778 (1960). 3. Stoker, J.J., "Water Waves," Interscience Publishers, Inc., New York (1957), pp. 187-196. 4, Yim, B., "Investigation of Gravity and Ventilation Effects in Water Entry of Thin Foils," Proceedings of the International Union of Theoretical and Applied Mechanics Symposium held in Leningrad (1971), NAUK Publishing House, Moscow, pp. 471-475 (1973). 5. Luke, J.C., "A Variational Principle for a Fluid with a Free Surface," Journal of Fluid Mechanics, Vol. 27, Part 2, pp. 395-397 (1967). 6. Gurtin, M.E., "Variational Principles for Linear Initial-Value Problems," Quarterly Journal of Applied Mathematics, Vol. 22, pp. 252-256 (1964). 7. Murray, J.C., "A Note on Some Variational Principles for a Class of Linear Initial-Boundary Value Problems," Journal of the Institute of Mathematics and Applications, Vol. 12, pp. 119-123 (1973). 8. Bai, K.J. and R. Yeung, "Numerical Solutions of Free-Surface Flow Problems," 10th Symposium on Naval Hydrodynamics, Office of Naval Research (1974). 9. Sao, K. et al., "On the Heaving Oscillation of a Circular Dock," Journal of the Society of Naval Architects of Japan, Vol. 130, pp. 121—=13F~ GVA”: dts} ae qe n\n, dip ed Shovel gore sue” ; a, “AOBRT). pais: - ne ¢ ae , 4 % wie ake al a "i a aT ' woh “xg aes, oy vies 6 pis ate “P dene kal Lewion niesnt aia ae, paniiossoxs ” Pik <9 984 .woovelt 43a iaebilnd Reg | i SMENRL) bos gdb at bt gd mieog® us eee OT SN 6) ie ENC j oe 7 nace 1 of =i rae ae . =< . : ; - the us De poatwe® gas? a ilate Grade & ro otqtoabsa\ te PatOd Dat Va! a ie ake hic st hea : ree sessde, qe <8 3 “4 aie fo oat set bait Wemedcer? SLB LRLS Set aya “i snes Ay bin eek tavern” ee ne Pera - el ra is * re eas 2) deer SES. SM — ; soy iad SEs naaljet Waking de, Ant Hof, pe i j KY ra " Pe) Per tie Os el a on: Kong’ * TPP GBEL Oe By sieht qantas ee 2 nel Ce ance re bys bre, 2d Otte a. etl MEAD ces Peentk Aires 7 Tigo Fo LBM o 2 =p td aptee USER coset ee ria Shel thee, yt: Sah. SE Th. 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