UNSTEADY LOW REYNOLDS NUMBER FLOW PAST TWO ROTATING CIRCULAR CYLINDERS BY A VORTEX METHOD

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1 Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference Jul 8-23, 999, San Francisco, California FEDSM99-8 UNSTEADY LOW REYNOLDS NUMBER FLOW PAST TWO ROTATING CIRCULAR CYLINDERS BY A VORTEX METHOD Masato Nakanishi Department of Energ Sstems Engineering Osaka Prefecture Universit Sakai, Osaka, Japan Teruhiko Kida 2 Department of Energ Sstems Engineering Osaka Prefecture Universit Sakai, Osaka, Japan ABSTRACT This paper treats the unstead flow generated b two circular clinders of equal radii which rotate with opposite angular velocities abruptl in flow initiall at rest. The stead flow of this problem, which is known as Jeffer s parado, was first studied theoreticall b Watson. He pointed out the default of earlier theories and showed that a uniform flow is generated at the outer field of the Stokes flow region from the clinders in which the flow is governed b the Navier-Stokes equations even if Renolds number based on the radius of the circular clinder and circumferential velocit is less than unit. However, he did not obtain the solution, since the outer flow is not solved. The main purpose of this paper is to know the mechanism of the generation of the uniform flow for the present problem. The vorte method used for numerical simulation is a vorte blob and sheet method and the cut-off function for the vorte blob is Gaussian. The viscous effect is simulated b random walk method. From the numerical results, we see that the near flow around two clinders is bounded in closed streamlines and the uniform flow is generated outside the closed streamlines. Further, the asmptotic analsis for the long time motion is carried out and it is shown that the outer flow is essentiall governed b the Oseen flow unlike the prediction of Watson. INTRODUCTION The problem of stead flow past rotating two circular clinders in a viscous fluid has long attracted to mathematicians, because it is impossible, in general, to obtain solutions of Stokes equations of slow viscous stead flow in Present address: Hitachi, Ltd., Mechanical Engineering Research Laborator, Tsuchiura, Ibaraki 3-3, Japan. 2 Gakuen-cho -, Sakai, Osaka , Japan. which the fluid velocit vanishes at infinit. Jeffer 922) solved the slow viscous stead flow generated b the rotation of two circular clinders b using the method which was published b him 92) on plane stress and strain in bipolar coordinates. If the clinders are outside one another, Jeffer found that it is impossible, in general, to make the fluid velocit vanish at infinit. He illustrated this b a detailed treatment of the case of equal clinders, rotating with equal speeds in opposite sense. Jeffer 922) used the bipolar coordinate sstem ξ,η): ζ = log{z + ia)/z ia)}, ζ = ξ + iη, z = + i. The curves ξ = constant and η = constant form sets of co-aial circles of the limiting point and the common point, respectivel, which are everwhere orthogonal. As, then ξ and η both tend to zero. In the Stokes flow the governing equation of the stream function ψ is the Biharmonic equation 4 ψ =, with ξ 2 <ξ<ξ, η π, and the no-slip condition on ξ = ξ and on ξ = ξ 2 is required. B converting this equation into a partial differential equation with constant coefficients for the dependent function Hψ, where H = cosh ξ) cos η))/a, we have the form: Hψ = f c ξ) cos nη) +f s ξ) sin nη), where the function f c,s ξ) satisfies a fourth order ordinar differential equation. Jeffer gave a suitable representation for the solution [see Elliott et al. 995) or Watson 995)]. For this epression, the uniform flow is generated at far-field and the velocit does not converge to zero at infinit. This is the Jeffer parado. To resolve this parado, Smith 99) obtained an Copright c 999 b ASME

2 asmptotic solution of the Stokes equations for the streamfunction which is valid at large distances from the clinders. This asmptotic epansion involves man unknown coefficients of the Fourier series and there was no obvious wa as to how these ma be obtained. Elliott et al. 995) established the boundar element method b using the asmptotic epansions given b Smith 99) and showed numericall that the combined bodies have no overall force or torque acting upon them. Watson 995) pointed out that the pressure field given b Smith s asmptotic form is not single-valued and proposed that the additional term to Jerrer s Fourier series of Hψ, sinh ξ log 2 cosh ξ 2 cos η) and sin η log 2 cosh ξ 2 cos η), is necessar. Further, Watson predicted that the outer flow is governed b the Navier- Stokes equations and its asmptotic epression is ψ C o log r + C )rsinθ as. However, he did not derive the force, since the outer flow which is governed b the Navier-Stokes equations is not obtained. Thus, the outer flow behavior is ver important in the stead flow problem. Furthermore, there is not an work on the unstead flow generated b two circular clinders of equal radii which rotate with equal angular velocit in opposite sense abruptl in flow initiall at rest. The main purpose of this paper is to know the mechanism of the generation of the uniform flow in the outer flow of the Stokes region and to discuss with the asmptotic form proposed in earlier theories. The boundar condition at far-field is important in this problem. Since the vorte method satisfies the far-field condition automaticall, this method ma be powerful, although the vorte method has been applied to various high Renolds number flows. In the present paper, the vorte method developed to a low Renolds number flow b the present authors Nakanishi et al. 997b) is used. THEORY OF UNSTEADY FLOW FOR LONG TIME MOTION An impulsivel started two-dimensional circular clinder with ver low speed is solved b an asmptotic approach in the previous work Nakanishi et al., 997a). In this asmptotic analsis, the show that for the long time motion the inner and outer flows are essentiall governed b the stead Stokes flow and the unstead Oseen flow, respectivel. Since in the present problem the velocit approaches to zero as, Watson 995) sas that the outer flow is governed b the Navier-Stokes equations. However, we here assume that the outer flow for long time motion is essentiall governed b the unstead Oseen flow in sense of the asmptotic analsis. We have to discuss this assumption after obtaining the outer solution. Figure shows the phsical coordinate in the present problem. The half of distance between the centers of circular clinders is defined as Ω Ω Figure. h Phsical coordinate h. The governing equation of the vorticit field ω is given from the Navier-Stokes equations: Dω/Dt = ν 2 ω. ) We take the tpical length as the radius of the upper circular clinder a and time is normalized with a 2 /ν. Then, Eq. ) is epressed as non-dimensional form: Dω/Dt =/Re) 2 ω, 2) where Re = Ω. Here, Ω is the non-dimensional angular velocit. From Eq.2), the vorticit field is epressed as 2 ω +2ε c ω ω ) =2εf, 3) t where ε = R e /2 and f =u + c) ω/ + v ω/. Here, c is some constant independent of time. We define: ˆζ = epεc)ω and ˆf = 2ε ep εc)f. Further, we define the Laplace transformation of these functions: Lˆζ) = ζ, L ˆf) = f and Lψ) = ψ. Then, we have ζ o ; p) = ε G o ; p) 2π f ; p)dd + F o, 4) D ψ o ; p) = ) ζ ep εc) log dd, 5) 2π o D where =, ), D is the whole flow region, S is the surface of the clinders. The function F o is defined b F o o ; p) = [ G s o ; p) ζ 2π S n 2 Copright c 999 b ASME

3 ζ ] n G s o ; p) ds. ) The function G is given b see Nakanishi et al., 997a) G ; p) =K a ), a 2 = εεc 2 +2p). 7) For the long time motion, p =2p/ε, the outer flow, ζ IV and ψ IV, is given b see Nakanishi et al 997a) ζ IV εα K c 2 + p) /2 R o ), 8) ψ IV 2π 2πε 2 Pf ep εcx) ζ IV log ρ R dxdy, 9) where X o = ε o, X o = R o cos θ o, sin θ o ) and ρ R = X X o. In these solutions, we assume that the order of f is smaller than the order of α with respect to ε. f = oα ). ) The asmptotic behavior of ψ IV for R o is given b ψ IV V α [ Ro εc 2 + p) /2 4 ) ] 2 R o log R o + A o R o where sin θ o, ) α 2π A o = Pf ep cx) ζ IV 2π c 2 + p) /2 sin θrdrdθ. 2) The first approimation of the inner solution for the long time motion is given b the stead Stokes flow, that is, Watson s solution. From the matching requirement, we can decide α and L in Watson s solution 995). α ΩA, c 2 + p) /2 sinh2α) E o + ce 3) Ω ΩL sinh2α) 4E o + ce ), 4) E = 2 γ + 4 log 4c 2 p 4c 2 log + c2 p c 2 + p ) ) + 4. ) We note that α is of the order of / log/ε). Therefore, if we decide c = O/ log/ε)), the assumption, Eq.), is satisfied. Here, we have from Eq.) ψ α c 2 + p) /2 4 + A o 2 log R o ) 2 sin θ o as R o. In the present analsis, we can not decide the value of c, however, we ma take c as the following value, since A o O/ log ε ), c = ΩA 2 sinh2α) log ε log2a) 2 T α)+. 7) 2 B using this result, the drag coefficient C D for t is given b see Watson 995 and Nakanishi et al., 997a) C D = 2π ) ep c 2 T o )S + cs 2, 8) R e h ct o where t =2T o /ε, and ep ) ep c 2 T o ) S = d, S 2 = d, B B 2 [ ao B = c ) c 2 log T o c log c T o + + ) c 2 log + ) ]2 + π 2 T o c 2, T o [ ao ] 2 B 2 = c + ) log ) + 2 log c + log + π 2 2. The asmptotic form, Eq.), sas that Watson s outer solution 995) is reasonable. Further, we see that the outer flow is essentiall governed b the Oseen flow unlike the prediction of Watson 995) and the deca of the drag force is of O ) T o log 2 T o as T o from Eq.8). where E o = 2 log ε log2a) T α), 5) 4 NUMERICAL METHODS The present paper aims to know the mechanism of the generation of the uniform flow numericall in Stokes region 3 Copright c 999 b ASME

4 a) t = b) t = Figure 2. Streamlines in the case of R e =and h =.5 and to discuss Jeffer s parado. In this problem, the main target is to know the outer flow generated b two circular clinders which rotate suddenl in flow initiall at rest. The governing vorticit equation in two-dimensional flow is given b Eq.). The velocit field is determined b the Biot Savart formula: u = K )ω)d, K ) 2, ) = 2π 2. 9) In the present vorte method, the fractional method is used for the evolution of vorticit: dω/dt =, dω/dt =/R e ) 2 ω. 2) The evolution of the vorticit field is calculated from the first equation of Eq.2) and the velocit field is solved b the panel method for a given vorticit field. In the present paper the panel method of source distribution given b delta function together with the constant vorticit distribution proposed b Kida et al.993) is used, under the flow tangenc condition. The vorticit is generated on the surface of the clinders such that no-slip condition is satisfied. In the present paper, the vorte sheet method proposed b Chorin 978) is used. The vorticit field is simulated b vorte blobs with Gaussian cut-off function. The viscous effect, the second equation of Eq.2), is simulated b the random walk method proposed b Chorin 973). The vorte sheet becomes vorte blob advected from a thin laer near the surface of the clinders to the outside of the laer. The thickness of this laer is of the order of /Re /2 for high Renolds number flow see Chorin 978)). However, in the present low Renolds number flow the boundar laer does not eist. In the previous work Nakanishi et al. 997b), this concept is shown to be available, provided that the thickness is taken as ver thin, because the flow near the surface of the clinders is almost along the surface, so that the generated vorticit flows almost along the surface even for low Renolds number flow. Therefore, in the present calculation, the thickness of the computational boundar laer is taken to be H = λ/re /2, λ =.3 in Re =. The vorte blob diffuses b viscous effect and in low Renolds number flow its velocit is large. Therefore, we have to take the small time step to increase the numerical accurac. In the present paper, the time step is where n is step number. t = ep n n +. ), NUMERICAL RESULTS The panel is constructed b dividing the polar angle equall and the number of panels is 8 on each circular clinder in the present calculation. The cut-off radius ε is taken to be l/π, where l is the panel length. Figure 2 shows the streamlines in the case of h =.5 and R e =. In these figures, the flow is almost smmetr independent of time. We see that the streamlines near the clinders are closed and a pair of the circulator flow is generated outside the 4 Copright c 999 b ASME

5 t =.2833 t = t = t = upper lower u 4 u t =.2833 t = Figure 4. The -component of velocit, u, on the -ais in the case of R e =and h = u Figure 3. The -component of velocit, u, on the -ais in the case of R e =and h =.5 closed streamlines. The generation of the circulator flow is also seen in the flow with impulsivel started circular clinder with constant speed see Nakanishi et al., 997a). This circulator flow induces a uniform flow in the outer field of the Stokes region. The rotation of a single circular clinder in high viscous flow generates the circulator flow whose tangential velocit is given b Ωa/r, where a is the radius of the clinder and r is the distance from the center of the clinder. The flow induced b the rotation of two clinders with opposite angular speed becomes circulator near the clinders, however, the induced whole flow can not pass through the gap between two circular clinders. Hence, the stagnation points eist on the -ais. Figure 3 shows the profile of the -component of velocit on the -ais. We see that the velocit of the upper and lower side of the clinders is almost proportional to /r with the development of time. The velocit in the gap becomes larger than the circumferential velocit of the clinder for the long time. Figure 4 shows the velocit profiles along the ais. This figure shows the remarkable change of the velocit near the gap. Further, this figure shows that the velocit increases with and becomes maimum at some distance, = O/R e ), and after then it becomes graduall zero with. Thus, we see that the Jeffer parado eists in the Stoke region. We see that the magnitude of the velocit becomes graduall maimum and its gradient is graduall small with. This feature implies that Smith s asmptotic solution is not reasonable. Figure 5 shows the time development of the drag and lift coefficients of the clinder, C D and C L. The suffi p and f denote the pressure and friction forces respectivel. We see that the friction drag is comparable with the pressure drag for R e = and the drag force becomes zero with the development of time, as predicted b Elliott et al.995). The lift coefficient is negative from the reason that the velocit in the gap is larger than the circumferential velocit, as shown in Fig.3, and the lift force due to the pressure is dominant. Figure shows the comparison with the present theor for the long time of motion. The global feature is almost the same. CONCLUSIONS The present paper treats the low Renolds number flow generated b two circular clinders of equal radii which rotates with opposite angular velocit abruptl in flow initiall at rest. For the stead flow of this problem, there is 5 Copright c 999 b ASME

6 4 C D C Df C Dp C D t