ACTA OCEANOGRAPHICA TAIWANICA SCIENCE REPORTS OF THE NATIONAL TAIWAN UNIVERSITY NO.3, PP , 4 FIGS, DECEMBER, 1973

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1 ACTA OCEANOGRAPHCA TAWANCA SCENCE REPORTS OF THE NATONAL TAWAN UNVERSTY NO.3, PP , 4 FGS, DECEMBER, 1973 THE RESONANT PHENOMENON NDUCED BY TWO OSCLLATNG DSKS N ROTATNG FLUDl CH-YUAN LEE2 ABSTRACT The flow induced by two infinite disks oscillating with frequency w about a state of steady rotation with angular velocity [} is analyzed. Assume the amplitude of oscillation is small and the viscosity is "low such that boundary layers exist. Linear theory shows that the Ekman-Stokes bound-, (E )1/2 ary layer is of thickness -2±w/fr on the top and bottom disks, where E is the Ekman number. The speclal.case of interest is where w/[}=2, i. e., the "resonant" case. n this situation the thickness of one boundary layer tends to become infiite, and so one must expect some strong effects well away from the disks. Using the solvability condition for the O(E) interior flow, we find that 0(1) interior flow (in contrast to O(E/2) interior flow in the non-resonant case) is induced by the oscillating disks. This 0(1) interior flow is an inertial balance with viscous vertical structure. The transitional period between "resonant" and "non-resonant" flow is also analyzed, which shows the growth of one boundary layer into an 0(1) interior flow with a time scale 0( -1). NTRODUCTON Oscillatory disturbances in a non-rotating fluid may be produced from viscous effects, for instance, a torsional oscillations of an infinite disk of frequency (J). The fluid is assumed to be homogeneous and incompressible with density p and kinematic viscosity v. Benney (1964) found that the solution consisted of an exponential decay in the circumferential velocity component within the Stokes layer of thickness 0(V/(J)1/2, and a steady first order inflow was produced in the far field. Benney (1965) also analyzed the flow field produced by rotational oscillations of a disk of frequency (J) about a steady rotation Q in a semi-infinite fluid also rotating with angular relocity Q. n this rotational case, we have two boundary layer lengths, i. e., the thicknesses O(V/(J)l/2 and 0(V/Q)1/2 of the Stokes and Ekman layers respectively, however, they arise in the combinations 0(v/12Q±(J) 1)11 2 The special case of interest is where (J)/Q=2, which we shall refer to as the "resonant" case, and the thickness of one of the boundary layers becomes infinite, thus large velocity must be expected well away from the plate. Wang (1972) solved the "non-resonant" case that a cylindrical tank, full of fluid, is oscillating with frequency (J) and rotating with angular velocity Q about its axis of symmetry, however, he failed to analyse the "resonant" case due to his neglect of the dramatic change of interior flow. n the present problem we consider the "resonant" case of the flow induced by two infinite disks separated by a finite distance L, oscillating with frequency (J) about a steady rotation Q. Using the solvability condition for O(E) problem, where E=DQ' we will find that 0(1) interior flow (in contrast to O(E/2) flow in the non-resonant case) is induced by the oscillating disks. This 0(1) interior flow is an inertial balance with viscous vertical structure and is the outcome of the "resonant" growth of one boundary layer. 1. Research done at Department of Mathematics, Clarkson College of Technology, N. Y., U.S. A. 2. nstitute of Oceanography, National Taiwan University, Taiwan. 77

2 78 Chi-YllcQ1l Lee FORMULATON Let us take a cylindrical coordinates (r, 0, z), rotating with angular velocity D, the corresponding perturbation velocity components are (u, v, w), r=o being the axis of symmetry, z=o, L being the two planes of the infinite disks (see fig. 1).?f =EV e 1LOt, T Ekman-stokes Boundary Layers. L 7f =EVBe i..mt o Fig. 1. SChematic of some of the physical dimensions in this problem. The equations governing the fluid motion are (ru)r+(rw). =0 VB Ut+Wr+wu.---2Qv=--p ( U) /1u-~r p r r B Ut+UVr+WV.+ ~ +2.Qu=(/1v--ia) 1 Wt+UWr+WW.= -pp.+v/1w (1) (2) (3) (4) (jb (jt 1 (j ",,here /1= (jrb- + (jzb +r Br ' These equations are to be solved subject to the followiq boundary conditions u=o ~t z=o.l w=o at z=o.l v=eyz~'''' at z-l v=ev.se'''' at z=o (5) ((i) (7) 00 where e is a small parameter corresponding to nonlinear effect, and VT and VB are 0(1) constants. t is consistent with the governing equations and boundary conditions to assume the following form of sowti9ns ll'prf. v=rg w=-2f p=+rh+~k (9) (10) (11) (12)

3 The ResfJCl1t Phenomenon nduced by Twa O$cillating Disks in Rotating Fluid 19 where F, G, H, and K are functions of z and t. The continuity equation is now automatically satisfied. ntroducing the dimensionless variables (denoted by primes) z=lz', t=w-1t', F=eLwF', G=eQG' H=epQwH', K=epQwK', w",qw' (13) Equations (2), (3), and (4) become (dropping the primes) F.t+e(F.2-2FF -G2)-2G= -H+EF... Gt +e(2f.g-2fg.)+2f. = EG.. r 2 Ft-2eFF.= 4LB H.+K.+EF where E= L~Q is Ekman number. Equation (16) implies H.=O, or H is only a function of t, and Ft-2eFF.=K.+EF with the boundary conditions F=O F.=O G=vTel..t G=vBel..t at z=o, 1 at z=o, 1 at z=1 at z=o where VT and VB are 0(1) constants. Assuming!~E~, we have the linear problem with viscous boundary layers Dear the two disks. Equations (14), (15) and (17) become F.t-2G=-H+EF... Gt+2F.=EG.. Ft=K.+EF.. Differentiating equations (22) and (23) with respect to z, we have F..t- 2G. = EF.. G.t+2F.. =EG.. (14) (15) (16) (17) (18) (19) (20) (21) Putting F..(z, t)==f..(z)el..t==f(z)el.. t and G.(z, t)==g.(z)et..t=g(z)et..t, equations (25) and (26) become iwf-2g=ef.. (27) iwg+2f=eg (28) NTEROR REGON (22) (23) (24) (25) (26) (A) The interior flow -; can be expanded as follows (Greenspan 1968) J:=ito+El/2it;.+Eli; +. 0(1) flow: iwfo-2g o=0 iwg;,+2f o=0 t is clear that the solution will be trivial unless uo ~21=0 2 'w (29) (30) (31)

4 80 Chi Yuan Lee or w=2 (assume frequency is a positive quantity). (a) w",,2 Fo=Go=O F o=a1z+aa Go=~Foz=~L w w where A 1 and As are arbitrary constants to be determined later. (b) w=2 F o = -io o =lo(z) where /o(z) is an unknown function to be determined later. (B) 0(E1/S) flow: iwf1-~=0 iwg 1 +2F 1 = 0 So the solutions for 0(E1/a) flow are similar to 0(1) flow. (C) O(E) flow: iwf a-2ga=/0.. This is the "resonant" case; The solutions are iwg s + 2F2= ilo (39) Nowwe will use the solvability condition for O(E) flow to determine the 0(1) unknow function /o(z) in the resonant case. (38)-i(39) and putting w=2, we have /0 =0 The solution is therefore, FO=C1Za+C2Z+CaZ+C4 Fo.=3C1z2+2Caz+Ca Go=iFo.=3iC1Z2+2iCaz+iCa (32) (33) (34) (35) (36) (37) (38) (40) (41) (42). (43) (44) where C h C a, C a, and C 4 are arbitrary constants, and will be determined in the viscous boundary layer analysis. BOUNDARY LAYER REGON t is evident that neither solution (32) nor (35) can satisfy all the boundary conditions, and the interior velocities have to be matched to the velocities of the solid surfaces within a boundary layer whose scale is sufficiently small so that the viscous forces become as important as the Coriolis force because of the large shears developed in the boundary layer. From the analyses of equations (27) and (28), the dimensionless boundary layer thickness is J=0(E1/S)=0(~) \ Q1/2L (45) and the natural coordinate scaling in the boundary layer is Z r;= 1/2- (46)

5 The Resonant Phenomenon nduced by Two Oscillating Disks in Rotating Fluid 81 Denoting the boundary layer velocities by prime "r", equations (27) and (28) become, to 0(1), iwfa'-2go'=f\h iwgo' +2Fo'-o.: introducing (48) into (47), we have (a) w::\:2: let Go-ea., then we have and G'O'l'lH -2iwG'O'l'l-{W S- 4)0 0' =0 a 4-2iwas +(4 - WS)=O (47) (48) (49) (50) w>2 w<2 Since the solutions have to remain finite for large 1/, only negative signs are chosen, Go'=B/e-a.l'l+Bs'e-a.sf (51) Fo' =ib/e-a.l'l-iba'e-a.s'l (52) where Bt' and lla'are arbitrary constants, and Go'= B 1e-"1" +Bse-a.s'J (53) F'0.=iB1e-a.l"-iBse-" s" (54) wt' = -2E1/sF o ' =2iE1/s(-.:B~{e-"'1" -1)- B s (e-a.s'j -1») (55) a1 as where B 1 and B s are arbitrary constants to be determined later. The boundary layer flow at Z= 1 is similar to the one at z=o. " (b) w=2: Equation (49) becomes G'0'J'J'J'-2iwGo'"" =0 Since the solutions have to remain finite for large 1/, we get (at z=o) Go' =A'e-a.l" +Co' - 1 Fo'=A'ie-"'ln- TiWCo' Where A' and Co' are arbitrary constants. Since Go' and F o ' also have to remain finite, we get Go'= Ae-a.l'1 F' o. = iae-"l'l 2iAE1/S w/ = (e-a.l'j -1) a l Similarly, the boundary layer velocities at Z= 1 are Go'=Be-"'le Fo.'=iBe-a.le, -2iBE1/2 ( -",' 1) W = e 10- al where ~=(-z)/el/s, A and B are arbitrary constants to be determined later. (56) (57) (58) (59) (60) (61) (62) (63) (64)

6 82 Chi Yuan Lee DETERMNATONS OF NTEROR AND BOUNDARY LAYER FLOWS Applying the boundary conditions and matching with the interior flow, we get, for boundary layer flow, (a) ()~2 () at z=o uo' =_i~b (e-"'l" _e-"'s")e Cwt (65) vo'0= ';B (e-cll" +e-clll")e'o>l (66) wo'=o (67) W' =iel/svl- -(e- Cll" -1)- (e-"'s" _1»)e Cwt (68) "al at (2) at z= ; uo'=~(e';''''le-e-clse)ecwt 'TV (69) v o ' 0= TV; (e-cl1e +e-clse)ecwt (70) wo'=o (71) wt' = _iel/svr( (e-clle-)- (e-cllle-»)ectdl al as (72) where the 0(1) interior flow is zero in this case (with A=As==O). We see that one boundary layer thickness _1_-+ 00 as (1)-+2. at section. (b) (/)=2: For interior flow: -ir uo=-2-[3(vb+vr)zs-2(2vb+vr)z+vs]e'w r vo=y(3(vs+vr)zs-2(2vs+vr)z+vs]ectd' wo=i(vb+vr)z8_(2vs+vr)z8+vsz]ectdt two typical examples are shown in fig. 2. This "resonant" case will be investigated in the following (73) (74) (75) -1 ~= 1 +1 ~=1 +1 (a) Anti-symmetric case Fig. 2. z=o ~:0.5 ~=O.211"--~---?ro(l) l=o (b) Symmetric case nterior flows in the resonant case; (a) v.s=-'r= 1 unit. (b) '.8='r=l unit. r=2,

7 The Resonant Phenomenon nduced by Two Oscillating Disks in Rotating Fluid 83 For boundary layer flow: () at z=o: (76) Wo'=o (77) (78) (79) (2) at z=l: (80) (81) (82) (83) TRANSTONAL PEROD BETWEEN "RESONANT" AND "NON-RESONANT" CASE Assuming the frequency m varies very slowly with time, we are interested in the response of the flow as m departs slightly from "resonant" frequency m=2. Put m=2+e:d1(t) (84) where f1(t) is an 0(1) known function and E:1 is a small parameter. The velocities are assumed as u, =[1 =F(E:s-1Z, E:at)e'''c v.= G;, = G(E:s-1Z,E:at)e'''' where E:z and E:a are two small parameters' get (85) (86) ntroducing (84), (85) and (86) into (25) and (26), we E:aFt+(im+iE:1fu)F-2G=E:s- zef.. (87) E:ic+(im+iE:fut)G+2F=E:s-zEG.. (88) (87)-i(88). we get E:aVt+iE:1Ui +htt)v=e:s-zev.. (89) where V=F-i&. Equation (89) is the transitional governing equation we wanted, The "nonresonant" regime corresponds to t-+oo or O( ;1 )and E:.-+E1/1, and equation (89) becomes ie:1httv=e:s-zev.. (90) Equation (90) is the Ekman-Stokes boundary layer governing equation. resonant frequency m=2, this corresponds to E:s-+O() and E:1~E, and f m is very close to the E:aV,=E:s-zEV.. (91) equation (91) implies E:a~O(E), i, e., the time scale near the "resonance" is 0(E-1). The transitional period is equivalent to E:a=E:1=E:s- se, and equation (89) becomes Vt+lt(t)V=V where!t(t)=.i(f1 +Jitt). The variation of the singular part of the Ekman-Stokes boundary layer flow and its corresponding governing equation are shown in fig. 3 and fig. 4. (92)

8 84 Chi-Yuan Lee co co co 1 7./(l{l '11,1 (~l non-resonant flow (a) Fig. 3. transitional flow (b) resonant flow (c) The singular part of the Ekman-stokes boundary layer flow during (a).non-resonantlperiod (b) transitional period (c) resonant period. (a) Fig. 4. The frequency variation and their governing equations corresponding to Fig 3. The boundary conditions at disks are V=V1(t) at z=o V =Va(t) at Z=1& Now we will use finite fourier sine transform to solve equation (92), define and V(n)= S: V(z)sin(nz)dz 2 co_ V(z) = - 1& "=1 and equation (92) becomes 2] V(n)sin(nz), O<z<1& n=,2,... Vt+N1(n.t)V=Na(n,t) (94) where N1(n,t)="z+h(t) and 1Va(n,t)=-n(V1+(-)"Va), and the solution of equation (94) is V =e-f~ Nd. (Co(n)+ J:Naef~ N 1 dt) (95) (93) where Co(n) will be determined by the. initial condition, i. e., the steady "resonant" this case. Thus (93) and (95) together will give the solution of equation (92). solution in DSCUSSON Since the time scale near the resonant frequency is O(E-1), it will be very difficult to verify the predicted 0(1) interior flow in the laboratory experiment. However, the Ekman-Stokes boundary layer expansion may be detected to some extent. f there is a lateral wall insteady of unbounded fluid in 'horizontaldirection; vertical boundary layers may not be possible, especially in the resonant case, and it is. worthy for further investigation. The analytic solution of transitional

9 The 如此棚 lilt Phenomenon nduced by two Oscillating Disks in :Rotω ing Fluid 85 governing equation may not provide useful information due to its slow convergence, thus numerical integration may be more convenient to analyse a specific problem. On the other hand, this resonant phenomenon may be analysed through nonlinear effect rather than viscous effect, however, it will not be discussed here. ACKNOWLEDGEMENTS would like to thank Professor A. C. Newell and M. J. Ablowitz at the Department of Mathematics, Clarkson College of Technology for their many valuable suggestions and helpful discussions. This research work was supported by the U. S. National Science Foundation. REFERENCES CTED BENNEY, 0.1. (1964) J. Fluid. Mech. 的 : 385. JlENNEY, 0.1. (1965) Quart. Jour. Mech. and Applied Math., 的 : 333. GREENSPAN, H. P, (1 96 的 The theory of rotating fluids, Cambridge Univ. Press, New York. WANG. CHANG-Y (1972) Cylindrical tank of fluid oscillating about a state of st 飽 dy rotation, Ph. D. Thesis, University of California at Los Angeles. 會轉流體之兵旅現象 字接運 摘要 有二無窮大之平根以角速度 Q 繞 - 共同軸旋轉, 同時以頻率 ω 在扭動 二平板間流體之運 動情形, 對海洋之循璟有其關連蝕 假定二平坡之振幅很小及流體之粘滯性很低, 線型理論證明 E V 月 Ekman-Stokes 邊層存在, 其單位寬度是 t 2±~ 刃有 r, E 是 Ekman 常數, 因此當 ω 20 時, 邊層即向外擴張至無窮遠, 此可稱為共振現象 利用高次流速有解答之必要條件, 吾人可發現單位為 1 之強烈內部流存在 ( 而言時日非共提時只有 單位為 E'/2 之內部流存在 此內部流係價性 (nterti 組 ) 平衡而有粘滯性之垂直結構 由非共振和 共振間之過渡期之分析, 可以發現此強烈內部流係一邊層流擴張而形成, 其形成之時間單位為 E- 0