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Simulations of Forces Acting on a Cylinder in Oscillatory Flow by Direct Calculation of the Nav~er-Stokes Equations T. Kinoshita Univertisty of Tokyo, Tokyo, Japan M. Hinatsu Ship Research Institute, Tokyo, Japan S. Murashige University of Tokyo, Tokyo, Japan Abstract Flow around floating vessels sometimes accompanies separations, and is unsteady in ocean waves. We select the flow around a circular cylinder in an oscillatory flow as a preliminary study for it. In this paper, the two dimensional Navier-Stokes equations for it are directly computed using body-fitted coordinate, moving mesh technique, 3rd order upwind scheme, and the MAC method at the Reynolds number, R-=10^ and the Keulegan-Carpenter number, K==5,7, and 10. The computed results simulate effect of the Kc number on the field in excellent agreement published data both qualitatively quantitatively. the flow with and Moreover, we examine the validity of this computational scheme. The good agreement with analytical solutions gives reliability of it, and we consider the effect of fineness of the grid and the effect of the Reynolds number minutely. 1. Introduction The flow around a circular cylinder in an oscillatory flow is idealization of the flow around floating vessel like semi-submersible in ocean waves. It accompanies separations and is unsteady. Hence, we must sometimes understand mechanism of the separated vortices to estimate hydrodynamic forces acting on it. Flow visualization gives a lot of valuable qualitative information for it. Williamson [1] oscillated a circular cylinder in water at rest at the Keulegan-Carpenter number, K=<60, and the frequency parameter, ~ =255, 313 where K==UmT/D, ~ =R-/K=, R." is the Reynolds number, R-=UmD/ v, Um denotes the maximum velocity in a period, D the diameter of the circular cylinder, v the kinematic viscosity of a fluid, and T the period of oscillation. He related the position and strength of vortices around it to K=. For example, a very interesting half Karman vortex street in the transverse direction was observed in the range of 7 OCR for page 313
expressed under the U=UmsinD as follows: Cd ~ 34 J F'~sin:9 do Cat ~ 2 UDT J F'n~,P,220 dig condition of (2) Sarpkaya[6] carried out an analysis of the experimental data of Keulegan and Carpenter[5] as well as his own data, and showed that the coefficients Cat and Cm depend on not only Kc but also R.-. In particular, it is interesting that Cat increases and Cm decreases with Kc in the range of about 51, may be reduced to Ca ~ 3K [ (ma) ~ + (ma) ~ - 4(~)~] (5) Cm = 2 + 4(~)l + (if)- (6) The equations (5) and (6) differ from (3) and (4) only in the last terms. All Stokes and Wang's solutions are solved. 314 virtually identical in the range of their validity, i.e. for large I. On the other hand, Bessho[9] obtained the analytical solutions for this problem using Oseen's scheme and showed the coefficients Cat and Cm as follows: Ca ~ 3~3 (~)-l [ 1 + ~ 15Kc ~ Cal ~ 2 + 4(~)~l t1 - O-8KC] (8) Numerical analysis is another way to this problem. Recent advances of a super computer have enabled direct calculation of the Navier-Stokes equations, and some flow fields have been solved. Baba and Miyata[10] carried out an analysis of a flow around a circular cylinder in an oscillatory flow at R-=1000 and K==5 and 7 using the finite difference method and showed qualitative agreement with observations. Objective of this paper is to accurately simulate instantaneous unsteady flow field around an oscillating circular cylinder and to estimate hydrodynamic forces acting on the circular cylinder quantitatively at R-=10000 and Kc=sm10 . Namely, increase of Cat and decrease of Cm wi th Kc in this range as well as drastic change of the flow pattern are simulated. For these purposes, the authors adopt direct simulation of the Navier-Stokes equations. Although the number of mesh is limited, we consider that simulation of newly generated predominant vortices which mainly affect the flow field is only required, while energy cascade to very small vortices is less important to this problem. In this paper, the Navier-Stokes equations are solved for flow around a sinusoidally oscillating circular cylinder in a fluid at rest, using the finite difference method. The flow is identical with flow around a circular cylinder fixed in an sinusoidally oscillating flow except for the constant gradient pressure (see Appendix). In the former case, we need the moving mesh technique which moves a computational grid with bodies, because the circular cylinder moves every moment. In addition to this -problem, the moving mesh technique enables computation of a flow around bodies which instantaneously change their shapes like fish. Accordingly, the present computational procedure can be widely applicable. problems are two-dimensionally Although three dimensional

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computations are necessary to take into account the three dimensional coherent structure of the shed vortex sheets, they remain as a next step of the research. This paper shows two dimensional solution to this problem. In order to use the present computational scheme as a practical tool, we must determine the validity of it. We compare the computed results with the analytical solutions and examine effects of total number of grid points, minimum grid size, time increment, and the Reynolds number on the computed results. 2. Governing Equations The two dimensional Navier-Stokes equations are written in the normalized form as follows: Kc at + ~ ax + v a - a0P + ~ ( a2U + a2 Kc at + u a6X + t; at = _ dP + 1 ( a2v + a2v) by Re ax2 au2 ( 9) where u and v denote the velocity in the x-direction and the velocity in the y-direction, respectively, and p the pressure. The velocity is normalized by the maximum velocity of oscillation in a period, Use, the pressure by P Um-2, the length by the diameter of a circular cylinder Do, and the time by the period of the oscillation T-. A superscript of asterisk denotes the dimensional value. ~ cut . . . \,, . -. . '.'; :.'.';',. ..\'k',''\\\ V ' ' . . , ~ \ \ . . . . . . . ~ . Fig.1 Grid circle. The number of grid divisions are 120 in the circumference direction and 50 in the radius direction. The grid is clustered near body surface using geometric series to obtain high resolution. The minimum space adjacent to the body is set to 0.005. Introducing a cut line along the positive part of the x-axis in the physical plane (x,y) as shown in Fig.1, we transform the plane (x,y) into the computational plane (hi) whose grid increment is set to be constant and unity for each direction as shown in Fig.2. J The continuity equation is written in the normalized form as follows: b o d y s u r f a c e au + an ~ o ax an (lo) 3. Computational Procedure 3.1 Body-fitted Coordinate System and Moving Mesh Body-fitted coordinate system makes it easy to compute flow around a body of arbitrary shape, especially to treat boundary conditions through coordinate transformation. In this simulation, the grid is generated as shown in Fig.1 and the system is an 'O-grid'. In this grid, a circular cylinder of unit diameter forms the inner boundary. The outer boundary is a circle whose diameter is 40 times that of the inner I- ~ I Fig.2 Computational plane . -r Actual computation is performed in this computational plane (Gil). As noted in the introduction, the moving mesh technique is adopted. Now we can consider two ways to move the grid as follows : (1) the way to fix the outer boundary and move only the inner boundary, (2) the way to move the outer boundary with the inner boundary simultaneously. Case (1) requires longer computational time than case (2), because the grid must be regenerated every moment. Hence, case (2) is adopted in this paper. The relations of partial differential 315

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operators between the physical plane (x,y) and the computational plane (hi) are written as follows: Ox ~ calf + be,, by ~ cat + d ~ ,, axe + ally ~ a a': + b ad + c a7,,, + day + e an t - - (am + CUT) a ~ - (bar + by:) ~ ~ + a t (11) where t denotes the time physical plane (x,y), ~ the the computational plane (Gil)' a - (T ~ JY7, . b - 77Z ~ -Jut c ~ (~ ~ - Jan' . d - ray ~ Jo a ~ a2 + c2, ~ - 2(ab + cd) c ab2 + ~ d ~ am + bar, + cc + do,, e - able + bb', + cat + dd,, J ~ ~ (~ (y ~ ~ I ~e ~~ I in the time in and All physical quantities, i.e. velocity and pressure, are estimated at each intersection point of grid, which is so-called 'regular mesh'. 3.2 Boundary Conditions Indices (i,j) of grid points in the computational plane(,) are shown in Fig.2. Each boundary condition is as follows: (1) Body Surface (j=1) *The velocity is set to that of a circular cylinder. *The first derivative of the pressure is derived through the Navier-Stokes equations (9) and the orthogonality of the grid on the body surface as follows: a rim ~ JEK (your - Eva) jEKC (W + do) (your - Eve) JE (W + dV) (yours - X:U,7) + R] E {-y~(cu~7, + eu,,) + x`(cv7,7, rev,,)} e (12) where U and V denote the motion velocity in the x-direction and that in the y-direction of the circular cylinder, respectively, and E ~ zi + ~ . 316 (2) Outer Boundary (j=J,J+1) *The velocity is set to zero. *Using linear extrapolation, the pres- sure is set as follows: pa = 2p_ - pa- 2 pare = 2p~ - pa_= (13) (3) Outside of Cut (i=-l,O,I,I+1) *The velocity and pressure are set as follows: q_ ~ = qua—2 q0 = qI—1 qI = ql q=-1 = q2 (14) where q denotes velocity or pressure. This condition is called 'periodic condition'. 3.3 MAC Method and Discretization In the present computation, the MAC method[11] is adopted as a computational algorithm. In this method, pressure p is obtained by solving the Poisson equation which is derived by taking the divergence of the Navier-Stokes equations as follows: V2p —V · { (U · V U — 1 as + 1 V2£ ) } Kc ~ t Re ~ 15) where u denotes the velocity vector and £ iS written as au av £ ~ a Sac + a (16) The condition of the continuity is imposed as follows: d£ A, _ £t (En ~ So) (17) where it denotes the time increment, and n the time step. The reason why the local dilation term of the n-th time step is left is that it prevents the accumulation of numerical errors. Velocity is obtained by solving the Navier-Stokes equations (9) using the Euler explicit time differencing scheme. The flow chart of computation is shown in Fig.3. The governing discretized as follows: equations are (1) Space differencing The terms except the convective term are discretized by 3-points central differencing scheme for all regions. The convective term is discretized by 1st order upwind differencing scheme (18) for j=2 and 3rd order upwind differencing scheme (19) for j> 3

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- ~ move body from (n)-th time step to (n+1)-th time step and set body surface ve- locity U=-l,v=-l . compute pressure p=41 by solving the Poisson eq.(15) . , ~ compute velocity until, v~-l by solving the Navier -Stokes eqs.(9) ! ~ + 11 Fig.3 Flow chart U( dd`)i ~ Ui (-Ui-l+ui+l) - luil (uil-2ui+ui+~) (18) ( d U ) (Ui-2 - 8ui l +8ui+1—Ui+2) i 4h (19) where h denotes the grid increment. (2) time differencing The Euler explicit method is used as follows: 61' ~ (U ~—Un) at fit (20) 3.4 Stability The limitation of the time increment is given by the von Neumann method, as follows: Kcxu (21) where fit denotes the time increment, ~x the local grid increment and u the local velocity. In the case of K==5 and u=1.0, the limitation of the time increment is given as follows: fit ~ 0.001 (22) Since the van Neumann method is valid only for linear equations, the 317 time increment must be set lower than the above value in actual computation. In this computation, it is set as follows: at ~ 0.0002 for Kc ~ 5 and 7 at ~ 0.~1 for Kc ~ to (23) 4. Computational Results The computations are performed at R-=10000 and K==5,7 and 10. In order to realize asymmetric flow field, sinusoidal motion of a circular cylinder in the direction vertical to the oscillation is superposed on the oscillation at the first quarter period of the oscillation as follows: U -sin(2~t) O.lxsin(4~t) for t ~ 0.0~0.25 0.0 for t > 0.2S (24) where U and V denotes the motion velocity of a circular cylinder in the x-direction and that in the y- direction, respectively. The amplitude and period of the superposed sinusoidal motion are 1/lOth and half of the oscillation, respectively. For all computational conditions, the circular cylinder is oscillated for eight periods, and the results are discussed for the data in the 8-th period. 4.1 Hydrodynamic Forces Acting on a Circular Cylinder (l)In-Line Force Published experimental data of hydrodynamic forces are arranged by the drag coefficient, Cat, and the inertia coefficient, Cm, or the added mass coefficient, C^, where Cm=l+C~ ( see Appendix). Before describing the computed results, we should note that there are scatters between the data published by different researchers as shown in Fig.4.1. The time history of in-line force shows a qualitatively same pattern in each cycle, but the magnitude and phase vary to some degree. The differences yield the scatters. For example, Kato et al.[12] showed the time history of pressure distributions, in which the magnitude differs more than 50 per cent in some cycle. Thus, since the value of the coefficients depends on how to average and estimate them, they may scatter even in the same experimental facility. Besides the differences of experimental method and conditions, the reasons of

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the differences of the magnitude and phase in each cycle are considered as follows. With progression of Rem and/or K=, irregularity of shed vortex sheet which has three dimensional coherent structures[2][3] is superposed on that which has two dimensional coherent structures. As a result, complicated flow field which shows strong irregularity is formed. Hence, the scatters are not only due to errors but also due to essential nature of flow field. Furthermore, the coefficients Cot and Cm (or C^) are considerably varied by even a little phase shift of hydrodynamic force time history. The computed results of Cat and Can are shown in Figs.4.1 and 2 with experimental data of Sarpkaya[6] and Tanaka et al.[13] and the analytical solutions[8][9] which are rewritten by equations (5) m(8) by noting Cm=l+C~. These figures show that these computational results simulate very well the effect of the Kc number on the drag coefficient Cat and the added mass coefficient Can. C d 25 2.0 15 10 05 \` 0.0 Exp. by Sarpkaya >I o Computed / o / /~ ,/' / / /~(Theory by Bessho / G>" " ~' Theory by Wang O ~ Fig.4.1 Drag coefficient, Cat C a 1.? 1.OI 08t 06: 04: 11 7 . . ~ 0.0 ~0.2 -04 ~ ~ ~ Theory by Wang O \ \ Theory by Bessho \? `( \ \% \ O \ Exp. by Sari\ o Computed \ -06 1 , , , , 1 1 4 6 8 10 12 14 16 K c Fig.4.2 Added mass coefficient, Ca Next, in order to examine that the present computation simulates unsteady phenomena accurately every moment, time histories of in-line force are shown in Figs.5.1,2 and 3. F in = F in, ~ p Um'2D ~2 ~ . ... 2. 5 F computed 2. an r- ________, MOrison ': / \ , ~ C) ~ - O 0, 5 , ~ ~ n _ / ~ \ / /, \ ~ I n s r— / I '' \\ - 1. C / , \' I ~ ~ - / I/ - 2 . n ~ ! , , , , ! , , . , ' , , , , ! , , , , ! An, to 1 n 2. ~ 3. C ¢. 0 5. 0 S. On 7 n Phase o= 2 ~ t -, T Fig.5.1 Time history of in-line force (K==5) F in = F in ( D U ~-2 D ·/ 2 ) . . , 2. ~ I ' ' ' — computed 1.5 ~ ---- Morison 0 1.^ c) O A, 5 u — w - fJ '~>' `) " n 7 \ ~ r ~ | \ ~ _ ~ c.s ~ ,1 'it' ~ H - ~ / ~ _ - !. ~ ~~ —2 . ~ ' ' ' ' I ~ ' , , , , ! , , , , 1 , , , , ~ , . . . n !. ^ 2. C 3. n ~ n s c s 0 7. n ase o= 2;~t T 4 6 8 lO 12 14 16 Fig.5.2 Time history of in-line force K c (K==7) \'` i no' a) - c,) 2.^ - S~ o . a) . - ,' i \\ ,! \ ~ /' ' . . ~ -` /, _ I ~ / H _, n ~ /,' ~ \ ~ , ~ I, , 2 . ~ x, , ! ~ , , , , I , . . ~ ^ . ^ ?. ~ ~ ^ Phase 318 Fig.5.3 Time history of in-line force (K==lO)

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Almost no experimental data of time histories of in-line force have been published, but we can get the approximation for them by substituting the experimental data of the coefficients into Cat and Cm in the Morison equation (1). The broken lines in these figures denote the approximation. Here it should be noted that the Morison equation assumes an odd-harmonic in-line force, neglects higher frequency components and cannot express the experimental data in the range of K==8m 25. The first assumption means that the flow field repeats every half period of the oscillation. At K== 5 and 7, the computed results are in very good agreement with the approximation by the Morison equation as shown in Figs.5.1 and 2. While at Kc=lO, they are not in so good agreement with them as at K==5 and 7 and are not odd-harmonic definitely as shown in Fig.5.3. This matter is discussed in Chapter 4.2 more minutely. (2)Transverse Force The comparison of the computed results of the maximum value of transverse force, F=~.m~x with experimental data of it[6] is shown in Fig.6. Ftranmax~ 3.0: 2 . 0 ~ . O 0 . 5 O. o o o 3 4 5 10 50 100 Kc Fig.6 Maximum value of Transverse force, F=~.m~x The experiment was not performed at R-=10000, but the computed results seem to show dependence of it on Kc and R." fairly well. The experiment shows that the ratio of the frequency of fluctuation of transverse force to the frequency of the oscillation of the cylinder is two in the region of the Kc number set in this computation. Time histories of transverse force in Figs.7.1,2 and 3 shows that. s — \ , \ ~ - I A ~ __ ! a) ~ ' V, SO \ / _ \ Fig.7.1 Time history of transverse force (K==5) 1 tran = F tran ~ p U m 2 D' 2 !. O I,, ,, ,, a) C) So O.v o a) -c. ~ U. a' ~ - ! . ~Q 5 - _, ED Exp. by Sarpkaya i. n !. n O Computed .,,,,, ., .. ~ 1 ' ' ' ' ~ ' ' ' . s ~ I \ A I \ 2. 0 3. 0 k. O Phase Fig.7.2 Time history of transverse force (K==7) F tran s F tran ~ p U ~2 D '' 2 ~ (. n , ,,, ,,,, I,,,, ,,,, i,, 3. 0 ~ 0) 2. C — O ,.c ~ 1 Lo - , an.. ~ \ ~ \ `¢ -2. ~ ~ \ i E-' - V -3. ~ — _ it. . o ~ I , , ! . , ~ i , ' , , , , ! , . , , ~ ^ ^ !. n 2. C 3. ~ (. n 5. ~ 5. n ?. n 0 = 2 Ir ~ T I \ An, 11 - - Phase Fig.7.3 Time history of transverse force (Kc=lO) 319 We consider that a transverse force depends on positions of vortices strongly. Time history of transverse force is a good measure of periodicity of the flow field.

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4.2 Time Evolution of Vorticity Time evolution of vorticity is shown in Figs.8.1 and 2 where the time increment is set to 0.1, contour pitch 2.0, a solid line denotes a counterclockwise vortex, and a broken line a clockwise vortex. The circular cylinder is oscillated from side to side along the x-axis, a black circle in each figure shows the position of an oscillating circular cylinder, and it is located on the left end at t=7.0. Williamson[1] classified aspects of vortices around the circular cylinder into some groups by Kc for R." of the order 103, and in the range of K=<13 they are summarized as follows: (a) K=<7 Vortices which are generated in a half period become a pair with vortices which are generated in the last half period. He called it 'pairing of at- tached vortices', although one of the pair is not attached but shed. (b) 71. When the Reynolds number is fixed to 10^, the valid range is Kc< OCR for page 313
(O) t=7.0 ·> (I) t=7.} (6) t=7.6 Fig.8.1 Vorticity contour, pitch=2.0, K==5 solid line: counterclockwise, broken line: clockwise 321

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~ rim ~~ f f ) . - ; r . A (3)t=7.3 D D' . (4)t=7.4 MA B_A ~ at\ B' , I] J // ,.. ,, ;... ,... , ;;,. i,. ;, '1 ~ D I !/ . · . / In_-, =. ~ . - , =.. · ' ,. B' I \ N ~ \ ~ (8jt=7.8 ~= A- ~ . ..... .~ . `:; . D ,, ~ me: i.... . . . "N Fig.8.2 Vorticity contour, pitch=2.0, Kc=lO ·. \., solid line: counterclockwise, broken line: clockwise 322 ., .. ~ ~ I/ — \ ., ~ <. E

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CCI 1.40 1.20 1.00 1 n Rn -l . _ 1 0.60 _ \ 0.40 _ 0.20 _ · compu ted Bes sho O. ~ ~` a) | - !. ~ ~ \ /~ Wang Kc Fig.9.1 Comparison of drag coefficient, Cat between computed results and analytical solutions Cm 2.10 _ 2.05 _ 2.00 - 1.95 _ 1.90 _ 1.85 _ 1.80 _ ~ · compel 0.00 1.00 2.00 3.00 4.00 5.00 Kc Fig.9.2 Comparison of inertia coef- ficient, Cm between computed results and analytical so- lutions U - sin(2~t) V; ~ O.Olxsin(4~t) for t - 0.0~0.25 ~ O.O for t ~ 0.2S (25) In equation (25), the amplitude of superposed sinusoidal motion in the y- direction is set to 1/lOth of equation (24) and the period the same as equation (24). There are no differences between the two results qualitatively. Time histories of in- line force are shown in Fig.10. Here it should be noted that it is difficult to get the same disturbance in different water tanks, even though same experiments are performed. This is one of the reasons why experimental data scatter. Although it is almost impossible to express the disturbance in the tank numerically, we consider F in = F in / (p Us'2D'/2 ) I,,0,,,,,.,,,,,,,,,.,,,,,,.,,~., ~ computed ( di s t xo . 1 ) ,~\ 3 0 t ~ , computed // \ r ~~~~~-~~ Morlson 1/ \ \ {, /. — — ' ",\ \' / J . 1,,,, 1 ., . . 1, . . . !. ~ 2. 0 3. 0 4. 0 5.0 6.0 7. ~ Phase 0= 2 ~ t ·/T - Fig.10 Effect of initial disturbance that the difference between computed results with the different initial disturbances as shown in Fig.10 may be obtained under different experimental conditions. (3) Effect of total number of grid points on Cat, Cam, and flow field The grid used in the present computation is determined by the total number and the minimum increment. Resolution of flow field by finer grid is higher, while stability of the solution is lower if the same time increment is used. Since we cannot compute so long, minimum time increment is limited. Hence, we cannot make the grid infinitely fine. Table 1 shows that Cut and Can computed by the grid 120x 50 and 140x 60 are in the range of scatters of experimental data, but that those by the grid lOOx 30 are clearly not in the range. Table 1 Effect of total number of grid points on Cat and C" (K==5, R.=104) r loo x 30 120 X50 140 X60 Cd 1.031 0.480 0.664 Ca 0.649 0.795 0.809 Figure 11 shows that the computed results by the grid 120 x50 and 140 x60 qualitatively agree with experimental data very well, but that those by the grid lOOx 30 do not agree with them. 323

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\\\\// ~ :: \~\~\`'t'\~\\\~'\~Al Il l l Ail ,l I/ // l, (8)t=7.8 (a) 100x 30 1 1 - ~ /~"f ',,' -/'~,1 (8)t=1.8 (b) 120x 50 (8)t=1.8 ~11~ Cal (c) 140x 60 Grid near the cylinder surface Vorticity contour Fig.11 Effect of total number of grid points on flow field (K==5, R.=10^, t=7.8) 324

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(4) Effect of minimum grid increment on Cat and C" We can do the same consideration as the total number of grid poets. Finer grid increment near the surface of the cylinder makes the resolution of boundary layer higher, but the stability of the solution lower if the same time increment is used. Table 2 shows that the coarser and finer minimum increments than that used in the present computation do not give good results of Cut and Cam. Table 2 Effect of minimum grid increment on Cal and C" (K==5, R-=104) x=0.003 ~ x=0.005 ~ x=0.007 0.703 0.480 0.703 ~ 0.469 0.795 0.469 (5) Effect of time increment on Cut and C - . Through the consideration of the stability, finer time increment makes the stability of the solution higher and give a solution closer to the true one. But finer time increment makes time derivative of the residual of the continuity equation larger and the results may not suffice for the mass conservation law. Table 3 shows that Cat and Can using ~t=O.OOO1 are not in the range of scatters of experimental data. Table 3 Effect of time increment on Cat and Can (K==5, R-=104) ; ~t=0.00041 1 1 --1 Cd Ca ; . 0.814 0.480 0.668 0.795 0.654 0.739 (6) Effect of the Reynolds number on Cat and Can The experimental data by Sarpkaya[6] and Tanaka et al.[13] indicate that Cat decreases and Can does not vary with progression of the Rue number. Figures 12.1 and 2 show that the computed results do not simulate the tendency. 3 CC 1.80 . 1.60 1.40 _ 1.20 _ 1 r,n _ lo lo O Computed Exp. by Tanaka et al 0 0 x 104 1.50 2.00 2.50 R. Fig.12.1 Effect of the Reynolds number on drag coefficint, Cut C,-. l.oor r, A r. L _ . ~ ~ 0.60 _ 0.40 _ 0 . 20 _ O Computed x104 0.50 1.00 1.50 2.00 2.50 O O Exp. by Sarpkaya R. Fig.12.2 Effect of the Reynolds number on added mass coefficint, Can Why can we get very good solutions at R-=10^ ? We can consider the reason as follows. As noted earlier, the big vortices generated every half period are predominant for the flow field. Hence, in the computation at R-=10^, we can consider that 3rd order upwind scheme controls the transition to turbulence near the surface of the cylinder and the flow field outside it, including predominant vortices, is correctly estimated. For this reason, we can obtain very good results at R-=104. Then we need to consider the effect of numerical dissipation on the solutions. At the higher Reynolds number, the local effective Reynolds number near the surface of the cylinder becomes too high for 3rd order upwind scheme to control the transition to turbulence there, and, as a result, the numerical dissipation makes the Reynolds number lower. he need to consider the cause of some errors in more detail to estimate the dependence of the Reynolds number on the flow

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field. For more precise estimate of hydrodynamic forces Acting upon a circular cylinder in an oscillatory flow, the following is hereafter required; *observations of phenomena for long periods *simulation of three dimensional coher- ent structures of the flow field. In this paper, all computation is performed using FACOM VP-100, and it takes about 70 minutes for 8 periods of the oscillation under the condition of the time increment ~t=0.0002 which is 40000 time steps. 6 Conclusions (1) Solutions for the flow around an oscillating circular cylinder at R-=10000 and K==5,7 and 10 are obtained by the direct calculation of the Navier-Stokes equations using body- fitted coordinates system, moving mesh technique, and the MAC method. The results are in excellent agreement with published experimental data quantita- tively. (2) The computed flow patterns show the 'pairing of attached vortices' and 'transverse street' at this range of K=, where Cat increases and Cm decreases drastically. The change of aspects of vortices is very similar to Williamson's observation at R." of the order 103. (3) Time histories of the computed in- line force are in good agreement with those which are given by the Morison equation and published experimental data of Cat and Cam at K==5 and 7. They show that this computation simulates unsteady phenomena accurately every moment. (4) Time history of the computed in- line force is not odd-harmonic at all at Kc=lO. (5) The reliability of the present computation is confirmed by the comparison of it with the analytical solutions. On the other hand, the computed results vary with total number of grid points, minimum grid increment, and time increment. But we cannot determine the best ones before getting solutions and comparing them with experimental data. In order to estimate The effect of the Reynolds number on the flow, we need to consider the cause of some errors in more detail. (6) For more precise estimate of hydrodynamic forces around an oscillating circular cylinder, structures of the flow field must be observed for long period and regularity of it must be investigated. Since three dimensional coherent structures of the flow field may affect the regularity, we need three dimensional simulation to discuss them more completely. The authors acknowledge some useful comments by Prof.Y.Ikeda of the University of Osaka Prefecture and Dr.Y.Kodama and Dr.T.Hino of Ship Research Institute. References 1 326 . Williamson, C.H.K., "Sinusoidal flow relative to circular cylinders", J. Fluid Mech. Vol.155, p.141 (1985) 2. Honji, H., "Streaked flow around an oscillating circular cylinder", J. Fluid Mech. Vol.107, p.509 (1981) 3. Tatsuno, M., and Bearman, P.W., "Flows induced by a cylinder per- forming oscillations at large ampli- tudes", J. the Flow Visualization So- ciety of Japan. Vol.8, No.30, p.357 (1988) 4. Morison, J.R. et al., "The force exerted by surface waves on piles", J. Petroleum Tech. Vol.189, No.2, p.149 (1950) 5. Keulegan, G.H., and Carpenter, L.H., "Forces on cylinders and plates in an oscillating fluid. J.Research of the National Bureau of Standards", Vol.60, No.5, p.423 (1958) 6. Sarpkaya, T., and Isaacson, M., "Mechanics of Wave Forces on Offshore Structures", Van Nostrand Reinhold Company (1981) 7. Stokes, G.G., "On the effect of the internal friction of fluids on the motion of pendulums", Trans. Camb. Phil. Soc. Vol.9, p.8 (1851) 8. Wang, C.Y., "On the high frequency oscillating viscous flows", J.Fluid Mech. Vol.32, p.35 (1968) 9. Bessho, M., "Study of viscous flow by Oseen's scheme(4th report, two- dimensional oscillating flow without uniform velocity", J. the Society of the Naval Architects of Japan. Vol. 161, p.42 (1987) 10. Baba, N., and Miyata, H., "High- order accurate difference solution of vortex generation from a circular cylinder in an oscillatory flow", J. Comput. Phys. Vol.69. No.2, p.362 (1987) 11. Harlow, F.H., and Welch, J.E., "Numerical calculation of time de- pendent viscous incompressible flow of fluid with free surface", The

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Physics of Fluids. Vol.8. No.12, p.2182 (1965) 12. Kato, S., and Ohmatsu, S., "Hydro- dynamic forces and pressure distribu- tions on a vertical cylinder oscil- lating in low frequencies", OMAE (1989) 13. Tanaka, N., Ikeda, Y., Himeno, Y., and Fukutomi, Y., "Experimental study on hydrodynamic viscous force acting on oscillating bluff body", J. the Kansai Society of Naval Architects. Vol.179, p.35 (1980) Appendix Relations Between Flow Around an Oscillating Circular Cylinder in a Fluid at Rest and Flow Around a Circular Cylinder Fixed in an Oscillating Flow (All quantities are dimensionally defined) (a) Flow around an Oscillating Circular Cylinder in a Fluid at Rest Assuming that a circular cylinder oscillates governing follows: dU + au, dt at in the x-direction, the equations are written as + u'aU. + tray _p apt + v(a u2 at + u ax. + v all' ~—ad P. + v( a V2 a y, 2 , +a2v ayes (26) au + av2 ~ 0 (27) where U=U(t) denotes the velocity of the circular cylinder, a superscript of prime the value in the coordinate system which moves with the circular cylinder, and v the kinematic viscosity. Boundary conditions are written as follows: u' = - U at r — v' = 0 u' = 0 v' = 0 at r ~ a where r denotes the distance from the center of the circular cylinder and a the radius of the circular cylinder. (b) Flow around a Circular Cylinder Fixed in an Oscillatory Flow Assuming that the ambient flow oscillates in the x-direction, the governing equations are written as follows: ant + U0U + viny eat + UbV+ any _ _ asp + v( ~ 2 + a 2) P a y ( a X2 a y2 ) (29) (30) where U=U(t) denotes the velocity of the oscillatory flow. Boundary conditions are written as follows: u = —U {v = 0 at r - u = 0 (v = 0 at r = a (31) In infinitely far field from the circular cylinder, the velocity is written as u = - U (t) v = 0 (32) Thus, equation (29) in the far field becomes the following equations: dt P ad (33.a) ~g (33.b) where Ping denotes the pressure in the far field, i.e. the pressure in the uniformly oscillatory flow. Equation (33.b) represents that the pressure in the oscillatory flow, Pins, is a function of only x and t, i.e. Ping = P,~(x,t). The pressure is divided into two parts as follows: (28) p = Pawn + Pa,-= (34) 327 where P=i-= denotes the pressure due to the disturbance caused by a circular cylinder. Using equations (33) and

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(34), equation follows: d t + ad t + U ad X + V Ha y (29) is rewritten as 1 apsis at p ~ ~ + v ( 2 at +ua~ +vag J + a 2) an 1 aPaist _ _. P dY ( a 2v Arc a2v at Equation (26) is the same form as equation (25). The only difference between the flow(a) and the flow(b) is that the pressure includes the additional term Pans whose gradient is required to accelerate the undisturbed flow in the flow(b). Integrated around the cylinder, Pawn gives rise to an inertia force, which is pa a 2) ~U/0 t per unit length. The inertia coefficient, Cm, which is defined in the flow(b), is related to the added mass coefficient, Can, which is defined in flow(a) as follows: (35) Can = 1 +- Cam (36) 3~