Added mass of a disc accelerating within a pipe J. D. Sherwood Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, United Kingdom H. A. Stone Division of Engineering & Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 ͑Received 1 April 1997; accepted 7 July 1997͒ The flow of inviscid fluid around a disc in a pipe is computed, and the results are used to determine the added mass of the accelerating disc in the frame in which the mixture velocity is zero. The added mass of an array of discs spaced at regular intervals along the pipe is then computed, and is related to the pressure gradient along the pipe. Some flow profiles are also presented. The results show that the added mass per particle increases as the pipe diameter is reduced relative to the particle size. The added mass per particle decreases as the number density of particles increases, but the added mass per unit length of the pipe nevertheless increases. Thus an increase of either the particle size or number density leads to a tighter coupling between the liquid and the particles; this result should hold for other particle shapes and configurations. Results are also presented for the drift, i.e., the displacement of fluid particles caused by the motion of an isolated disc along the axis of the pipe. If the diameter of the pipe is sufficiently small, the added mass of the disc is modified from that in unbounded fluid, and the background drift at the walls of the pipe can no longer be estimated from the added mass of the disc. © 1997 American Institute of Physics. ͓S1070-6631͑97͒02511-7͔
I. INTRODUCTION Published results for the added mass coefficient Ca vary slightly, since some authors assume that all bubbles have the Models of two-phase flow require a constitutive relation same velocity whereas others allow bubble velocities to vary. for the forces which act between the liquid and gas. An ap- However, Sangani et al.3,7 concluded that results of their nu- proximation sometimes employed divides the forces into a 1–3 merical solutions of the full equations were closely approxi- sum of drag and added mass terms. The added mass of a mated by the expression single spherical bubble in unbounded irrotational flow is 4 well-known, and results are available for the added mass of 1ϩ2 a spherical bubble in fluid which is accelerating.4–6 The ra- C ϭ , ͑3͒ a 1Ϫ dius of the bubble need not be constant.5 However, few re- sults are available for the added mass of non-dilute suspen- obtained by Zuber8 using a cell model. Ishii et al.9 performed sions, and fewer still treat flow in bounded geometries. We time-dependent computations of bubbly flows assuming an treat analytically an isolated disc accelerating in a pipe, as 10 added mass Caϭ1ϩ2.78 due to Van Wijngaarden. Thus well as a regular array of accelerating discs, in order to ob- there is a consensus of opinion that added mass increases as tain more insight into this problem. the volume fraction of liquid decreases ͑assuming that phase A review of added mass in non-dilute suspensions is 3 inversion does not occur͒. This increase can be thought of in presented by Sangani et al., who are concerned with small terms of a tighter coupling between the dispersed particles amplitude oscillatory motion of bubbles. The average hydro- and the liquid. dynamic force ͗F͘ per bubble can be written in the form Many multiphase flows are confined to pipes, and only a few analytic results are available for such flows. Smythe 1 ˙ ˙ ˙ ͗F͘ϭ 2Cavb͗umϪv͘ϩvb͗um͘ϩ12RCd͗umϪv͘, computed the streamfunctions for inviscid irrotational flow ͑1͒ in a pipe containing either a sphere11 or a spheroid12 ͑includ- 13,14 where and are the density and viscosity of the liquid, R ing the limiting case of a disc͒. Cai and Wallis extended and v are the radius and volume of the bubble, u is the this analysis to linear arrays of spheres and obtained added b m 14 mixture velocity, v is the bubble velocity, and ͗.͘ denotes an mass coefficients; they also reported the added mass of a single disc accelerating broadside in a pipe. We reconsider averaged quantity. The added mass coefficient Ca , and vis- the case of a disc, using methods based on the velocity po- cous drag coefficient Cd , have been normalized so that they approach unity as the volume fraction of bubbles  0. tential, and then study a linear array of discs in a pipe, work- Equation ͑1͒ can alternatively be written in terms of the→ av- ing in a frame in which the average mixture velocity of the erage liquid velocity ͗u͘ using the relation fluid and particles is zero. Acceleration of the particles leads to acceleration of the surrounding fluid, and this acceleration
͗um͘ϭ͑1Ϫ͒͗u͘ϩ͗v͘ ͑2͒ reaction, which corresponds to the added mass, creates pres- sure gradients. The net result is a macroscopic pressure gra- if the volume fraction of bubbles, , is assumed constant. dient along the pipe. We find that the added mass coefficient From now on we neglect the viscous drag term, and set the Ca ͑per particle͒ increases as the particle size becomes closer viscosity ϭ0. to that of the pipe. As the particle number density increases,
Phys. Fluids 9 (11), November 1997 1070-6631/97/9(11)/3141/8/$10.00 © 1997 American Institute of Physics 3141
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp ϱ
͑r,z͒ϭ bnJ0͑nr͒exp͑Ϫnz͒, ͑7͒ n͚ϭ1
where the n are chosen to satisfy
J1͑na͒ϭ0, ͑8͒
FIG. 1. A single disc, of radius R, on the centreline of a pipe of radius aR. thereby ensuring that boundary condition ͑6c͒ is satisfied. Boundary conditions ͑6a,b͒ lead to ϱ the added mass per particle decreases, because of shielding, Ϫ1ϭ bnnJ0͑nr͒,0рrϽ1, ͑9a͒ n͚ϭ1 but the added mass per unit length of pipe always increases. As an additional application of the analysis, we calculate ϱ the drift of fluid markers disturbed by a steadily translating 0ϭ ͚ bnnJ1͑nr͒,1рrрa. ͑9b͒ disc in a pipe. We are thus able to study the influence of the nϭ1 pipe walls on the typical magnitude of marker displacement. In order to treat these dual series equations, we follow the method of Cooke and Tranter,15 as described by Sneddon.16 For completeness we present the most important intermedi- II. ANALYSIS ate results, beginning with the identity ϱ We first consider the velocity potential for steady invis- Jϩ2mϩ1Ϫp͑n͒J͑nr͒ ϭ0, 1Ͻrрa, ͑10͒ cid flow around a single disc of radius R moving broadside ͚ 1Ϫp 2 nϭ1 J ϩ ͑na͒ with velocity U along the axis of a pipe of radius Ra,as n 1 1 1 depicted in figure 1; results, calculated using a different where Ϫ 2рpр 2, ϾpϪ1 and the n are the roots of 14 method, have been reported by Cai and Wallis. In Sec. III, J(an)ϭ0. The boundary condition represented by ͑9b͒ we show how the added mass of an accelerating disc may be may be satisfied if we take ϭ1 and evaluated once the velocity potential is known, and numeri- ϱ J ͑ ͒ cal results are presented in Sec. IV. In Sec. V we consider a 2mϩ2Ϫp n nbnϭ cm Ϫ , ͑11͒ ͚ϭ 1 p 2 one-dimensional array of discs placed at regular intervals m 0 n J2͑na͒ along the axis of the pipe. The discs have zero thickness, so i.e., that the volume fraction of discs within the pipe is always ϱ zero, even though the particle number density is non-zero. 2Ϫp 2 Ϫ1 bnϭ͓n J2͑na͔͒ cmJ2mϩ2Ϫp͑n͒. ͑12͒ Nevertheless, a suspension of larger and more numerous m͚ϭ0 discs ought to correspond physically to bubbly flows with increasing volume fraction of bubbles. Finally, fluid trans- An expression for the unknown cm is obtained by substitut- port, studied via the concept of drift, is discussed for this ing ͑11͒ into ͑9a͒: bounded system in Sec. VI. ϱ ϱ c J ͑ ͒J ͑ r͒ We assume an irrotational, incompressible flow, with ve- m 2mϩ2Ϫp n 0 n Ϫ1ϭ Ϫ ,0рrр1. ͚ϭ ͚ϭ 1 p 2 locity n 1 m 0 n J2͑na͒ ͑13͒ uϭ“, ͑4͒ Some relations concerning Bessel functions are now required where the potential satisfies Laplace’s equation, in order to enable us to eliminate the r-dependence of ͑13͒. 16 2ϭ0. ͑5͒ From Sneddon ͑equations 2.1.19 and 2.1.20ٌ͒ ϱ Lengths are non-dimensionalized by R, and velocities by U. 1Ϫk u Jϩ2mϩk͑u͒J͑ru͒du We use cylindrical coordinates, with rϭ0 as the axis of the ͵0 pipe and the disc, and with the disc in the plane zϭ0. The boundary conditions, assuming no flow at infinity, are ⌫͑ϩmϩ1͒r͑1Ϫr2͒kϪ1 ϭ ͑kϩ,ϩ1;r2͒, kϪ1 F m 2 ⌫͑ϩ1͒⌫͑mϩk͒ץ uzϭ ϭ1, 0рrϽ1, zϭ0, ͑6a͒ z 0рrϽ1ץ ϭ0, rу1, ͑14͒ץ u ϭ ϭ0, 1рrрa, zϭ0, ͑6b͒ rץ r where F m(a,b;x)ϭ2F1(Ϫm,aϩm;b;x) is the Jacobi poly- . nomialץ urϭ ϭ0, rϭa, all z, ͑6c͒ ¯ r If we define the Hankel transform f (p)off(r)byץ ϱ ¯ 0, z Ϯϱ. ͑6dٌ͒ → → f͑p͒ϭ͵rf͑r͒Jn͑pr͒dr, ͑15͒ We consider the region zу0, and look for a solution of the 0 form then the inverse transform is
3142 Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp ϱ ⌽ ϭϪ , ͑25͒ ¯ z 0 f ͑r͒ϭ pf͑p͒Jn͑pr͒dp, ͑16͒ ͵0 where the subscript z indicates that the potential corresponds and the Hankel inversion theorem applied to ͑14͒ gives to motion of the disc in the z direction. Note that ⌽z Ϫ as z ϱ, and ⌽ ϭ0 over the plane external to the → 0 → z 1 ϩmϩ1 r1ϩ 1Ϫr2 kϪ1 disc. It is clear, by symmetry, that ⌽ (r,Ϫz)ϭϪ⌽ (r,z). Ϫk ⌫͑ ͒ ͑ ͒ z z u Jϩ2mϩk͑u͒ϭ ͵0 2kϪ1⌫͑ϩ1͒⌫͑mϩk͒
2 III. ADDED MASS ϫJ͑ru͒F m͑kϩ,ϩ1;r ͒dr. ͑17͒ The added mass of an object moving in unbounded fluid We also require the orthogonality relation for Jacobi polyno- at rest at infinity is discussed by Batchelor.4 We write the mials ͑see, e.g., Magnus et al.17͒, ͑dimensional͒ velocity potential ˜ in the form 1 2ϩ1 2 kϪ1 2 ˜ r ͑1Ϫr ͒ F n͑kϩ,ϩ1;r ͒dr ϭRU͑t͒ ⌽͑xϪx0͑t͒͒, ͑26͒ ͵0 • where x0(t) is the instantaneous position of the center of the ⌫͑ϩ1͒⌫͑k͒ body. In the absence of gravitational or other body forces, ϭ ␦ . ͑18͒ 2⌫͑ϩkϩ1͒ 0n the unsteady form of Bernoulli’s equation ͑equation 6.2.5 of Batchelor4͒ takes the form We now multiply both sides of ͑13͒ by r(1Ϫr2)Ϫp 2 ˜ ˜u2 pץ ϫF j(1Ϫp,1;r ) and integrate to obtain ϩ ϩ ϭC, ͑27͒ t 2 ץ 1Ϫp͒͑⌫ Ϫ ␦0 j 2⌫͑2Ϫp͒ where ˜u is the ͑dimensional͒ fluid velocity, and C is a con- ϱ ϱ stant, independent of position, if the flow is irrotational. c J ͑ ͒J ͑ ͒⌫͑ jϩ1Ϫp͒ m 2mϩ2Ϫp n 2 jϩ1Ϫp n Let S be the surface of the body, with outward facing ϭ Ϫ . 1 ͚ϭ ͚ϭ p 2 2p 2 4 n 1 m 0 2 ⌫͑ jϩ1͒n J2͑na͒ normal n. Again following Batchelor ͑p. 404͒, the hydrody- ͑19͒ namic force F acting on the body in an inviscid flow is ˜ ˜2 uץ Hence the cm satisfy the linear equations FϭϪ pn dSϭ ϩ n dSϪ Cn dS t 2 ͬ ͵Sץ ͫ pϪ1 ͵S ͵S 2 ⌫͑ jϩ1͒⌫͑1Ϫp͒ 1 1 1 Ϫ ␦ ͑28a͒ ⌫͑2Ϫp͒⌫͑ jϩ1Ϫp͒ 0 j
ϱ ϱ ϭR ͑U˙ ⌽͒n dSϩ ͓ 1˜u 2ϪU ˜u͔n dS, ͑28b͒ J2mϩ2Ϫp͑n͒J2 jϩ1Ϫp͑n͒ ͵ – ͵ 2 – ϭ c , 20 S1 S1 ͚ m ͚ 2Ϫ2p 2 ͑ ͒ mϭ0 nϭ1 J ͑ a͒ n 2 n since in ͑28a͒ the integral of C over the surface of the body i.e., is zero. The second term on the right-hand side of ͑28b͒ ϱ corresponds to the force in steady motion, and F–U can be shown to be zero either for steady motion in unbounded ͚ A jmcmϭBj , jϭ0,1,2, ..., ͑21͒ mϭ0 fluid, or for motion parallel to the axis of a pipe. The first where term in ͑28b͒ is the acceleration reaction, G. In the absence of viscous effects G accounts for the total hydrodynamic ϱ J2mϩ2Ϫp͑n͒J2jϩ1Ϫp͑n͒ force acting on the body as fluid surrounding the particle is A ϭ 22 jm ͚ 2Ϫ2p 2 ͑ ͒ accelerated. This acceleration reaction is often called the nϭ1 J ͑ a͒ n 2 n added mass of the particle. and If a disc of unit radius moves broadside with velocity U 18,19 pϪ1 in unbounded fluid, the velocity potential is B jϭϪ2 /⌫͑2Ϫp͒, jϭ0, 2UR ϭ0 otherwise. ͑23͒ ˜͑r,z͒ϭ ͑z cotϪ1 Ϫ͒ ͑0рр1, 0рϽϱ͒, The velocity potential (r,z) is constant over the plane z ͑29͒ ϭ0, 1рrрa external to the disc. We can evaluate this con- where the oblate spheroidal coordinates and are defined stant 0 at an individual point, but it is more convenient to by take the average value zϭ, r2ϭ͑1ϩ2͒͑1Ϫ2͒. ͑30͒ 1 a ϩ 0ϭ 2 ͑r,0͒ 2rdr The region ϭ0, 0рр1 represents the surface zϭ0 of ͑a Ϫ1͒ ͵1 the disc, on which ˜ϭϪ2UR(1Ϫr2)1/2/. The reaction on ϱ a disc of radius R accelerating in the z direction is 2 bnJ1͑n͒ ϭϪ 2 ͚ , ͑24͒ a Ϫ1nϭ1 n 4 U˙ R3 1 8 U˙ R3 2 1/2 GzϭϪ ͑1Ϫr ͒ 2rdrϭϪ , ͑31͒ and to work with the velocity potential ͵0 3
Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone 3143
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp as obtained by Lamb18 ͑p. 134͒. If the disc accelerates in unbounded fluid, the pressure at infinity is uniform, but if the disc accelerates within a pipe, the pressures far upstream and far downstream of the disc differ. We assume that p p1 as z ϱ, and that p p2 as z Ϫϱ. Evaluating Bernoulli’s→ equation→ ͑27͒ at →zϭϮϱ gives→
p1 p2 ϪRU˙ ϭ ϩRU˙ , ͑32͒ 0 0 and hence
p1Ϫp2 RU˙ ϭ . ͑33͒ 0 2
The acceleration reaction for the disc in a pipe is ˆ FIG. 2. The non-dimensional added mass coefficient Gz of a single disc in ϱ a pipe of radius a. ͑a͒ Numerical results; ͑b͒ asymptote ͑45͒ for aϪ1Ӷ1, 1 ˆ ˙ 3 ͑c͒ limiting value Gzϭ8/3 for a disc in unbounded fluid, ͑d͒ ᭝ non- Gzϭ2UR ͚ bnJ0͑nr͒Ϫ0 2rdr 13 ͵0 ͫ nϭ1 ͬ dimensional added mass of a single sphere in a pipe. ϱ a2 b J ͑ ͒ ˙ 3 n 1 n ϭ4UR 2 ͚ , ͑34͒ ͩa Ϫ1ͪnϭ1 n small for large. To avoid numerical underflow these func- tions are set to zero by S17DEF. The function in 23 was and the jump in pressure between zϭϮϱ is ⌫ ͑ ͒ evaluated by means of the NAG routine S14AAF. All the ˙ ϱ 4RU bnJ1͑n͒ infinite sums are approximated by finite sums, so that ͑7͒ p Ϫp ϭ2RU˙ ϭϪ 1 2 0 2 ͚ becomes a Ϫ1nϭ1 n N Gz ͑r,z͒ϭ b J ͑ r͒exp͑Ϫ z͒, ͑39͒ ϭϪ . ͑35͒ ͚ n 0 n n a2R2 nϭ1 Numerical results will be presented for the dimensionless where reaction MϪ1 b ϭ͓2ϪpJ2͑ a͔͒Ϫ1 c J ͑ ͒, ˆ ˙ 3 n n 2 n ͚ m 2mϩ2Ϫp n GzϭϪGz /UR . ͑36͒ mϭ0
The acceleration reaction Gz is in the opposite direction to nϭ1,...,N. ͑40͒ the acceleration U˙ : the change in sign in ͑36͒ ensures that The cm are obtained by solving the linear equations Gˆ Ͼ0. z MϪ1 Smythe12 and Sangani et al.3 discuss the relation be- A jmcmϭBj , jϭ0,...,JϪ1, ͑41͒ tween computations of added mass and of an analogous elec- m͚ϭ0 trical conductivity problem. Suppose the pipe is filled with conducting fluid, so that, in the absence of a disc, the elec- where trical potential is ϭVRz. If an insulating disc of radius R is N J2mϩ2Ϫp͑n͒J2jϩ1Ϫp͑n͒ inserted at zϭ0, and the total current remains unchanged, the A jmϭ Ϫ . ͑42͒ ͚ϭ 2 2p 2 electrical potential becomes n 1 n J2͑na͒ Equation ͑41͒ represents a set of J equations in M un- ϭVRzϪVR⌽z . ͑37͒ knowns. In practice, if we fix N and take MϭJ, ͑41͒ may be Hence the additional voltage drop ⌬V between zϭϮϱ due solved only for sufficiently small M: the maximum M for to the presence of the disc is which ͑41͒ may be solved by the NAG routine F04ATF de- pends upon the dimensionless pipe radius a. For the case a ⌬Vϭ2VR0 , ͑38͒ ϭ5.0, if we take pϭ0.5 and Nϭ1000, then Gˆ ϭ2.79 for which corresponds to the resistance of a fluid-filled pipe of z Mϭ112 and the matrix A is ill-conditioned for larger values length 2 R. 0 of M.IfNis increased to 2000, the corresponding values are ˆ Mϭ156 and Gzϭ2.73, whereas at Nϭ5000, Mϭ248 and IV. NUMERICAL RESULTS ˆ Gzϭ2.70. Results were similar for other values of p: p ˆ All Bessel functions were evaluated in double precision ϭϪ0.5, Nϭ5000 leads to Gzϭ2.70 for Mϭ248, with A using the NAG routines S17DEF, S17AEF and S17AFF, and ill-conditioned for MϾ255. ˆ roots of ͑8͒ were obtained using part of a package of Bessel Figure 2 shows values of Gz as a function of a, which function integration routines.20,21 Note that, for any given agree with those of Cai and Wallis.14 All results were com- value of the argument z, the Bessel functions J(z) become puted with Nϭ5000, pϭ0.5 and with M equal either to 300,
3144 Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp or to the maximum value for which equation ͑41͒ could be ˆ solved numerically. Gz decreases to 2.7 as a increases from 1 to 6. The numerical predictions then increase slowly as a increases further, rather than decreasing towards the value 8/3 predicted by ͑31͒. When aϭ50 the matrix A is ill-posed for MϾ80. Thus the numerical scheme becomes less satis- factory as a increases. When aϪ1Ӷ1, the flow into the narrow slit between the pipe and the edge of the disc may be approximated as that due to a line sink, restricted to an angle of /2. The total ͑non-dimensional͒ volume flux due to the motion of the disc at unit velocity is , and this flows into a slit of length (aϩ1), where we have taken the average of the inner and outer radii. Hence, close to the slit, the velocity potential has the form FIG. 3. The axial velocity profile uz in a pipe of radius aϭ1.5 at positions 2 zϭ ͑a͒ 0.01, ͑b͒ 0.1, ͑c͒ 0.2, ͑d͒ 0.5, ͑e͒ 1.0. ϭ ln r , ͑43͒ ͑aϩ1͒ s z 0.01 for the velocities presented here. Even so, results where rs is a radial coordinate local to the slit. If we assume у were poor on rϭ0, where we cannot rely on the decay of that ͑43͒ holds from rsϳaϪ1 out to rsϭ1, we might expect J0(nr)asn ϱ. Figures 3 and 4 show profiles of uz(r)at 2ln͑aϪ1͒ various axial positions→ z. Near the edge of the disc the po- 0ϳ , ͑44͒ Ϫ1/2 ͑aϩ1͒ tential gradient ٌ has an O(re ) singularity, where re is a local cylindrical radial coordinate.19 Hence it is not surpris- and hence, by ͑35͒, the acceleration reaction should vary as ing that the numerical scheme has difficulty in resolving the 4a2 ln͑aϪ1͒ velocity field in this region. Gˆ ϭ2a2 ϳ . ͑45͒ z 0 aϩ1 If the fluid velocity around a moving body is known, the viscous dissipation within the fluid may be computed, This asymptote ͑45͒ is shown in figure 2, and appears to be thereby giving an estimate of the viscous drag D on the body satisfactory. Quantitative agreement is improved if a con- for cases in which viscous dissipation in boundary layers is stant 1.4 is subtracted from ͑45͒. small ͑such as clean bubbles͒. For example, if separation Also shown on figure 2 are results for the acceleration does not occur and flow is irrotational, the potential flow reaction on a sphere in a pipe, which was shown by Cai and field may be used for such an estimate at high Reynold num- 13 Wallis to be bers. For a spherical bubble of radius R, moving at velocity U in unbounded fluid with viscosity , the drag is found4 to Gsphereϭ 4R3U˙ ͑CSmytheϪ1͒, z 3 0 be Dϭ12RU. However, for the case considered here the Smythe 11,12 ˜ 1/2 where C0 is a coefficient computed by Smythe. The velocity potential has an O(re ) singularity at the edge of acceleration reaction on a sphere in unbounded fluid is the disc: this can most easily be seen in the case of a disc 2R3U˙ /3, and hence, with the non-dimensionalization moving in unbounded fluid ͑29͒. Rates of strain are ˆ sphere O(rϪ3/2), and hence the dissipation at the edge of the disc is ͑36͒ adopted here, Gz 2/3 as a ϱ. The acceleration e reaction on a sphere increases→ much more→ rapidly than that on a disc as the diameter of the pipe approaches that of the particle. In the case of a translating disc, rapid flow occurs only in a singular region around the edge of the disc. In the case of a sphere, rapid flow occurs along the entire length of the slowly varying narrow gap between the sphere and pipe.13 We may also compute the velocity field, given by
ϱ
uz͑r,z͒ϭϪ bnnJ0͑nr͒exp͑Ϫnz͒, ͑46a͒ n͚ϭ1
ϱ
ur͑r,z͒ϭϪ bnnJ1͑nr͒exp͑Ϫnz͒. ͑46b͒ n͚ϭ1 We might expect that ͑46a,b͒ should converge, for reason- able bn , when zϾ0. However, convergence appears to be more problematic on zϭ0. In practice, computations of ve- FIG. 4. The axial velocity profile uz in a pipe of radius aϭ4.0 at positions locity are best performed for z slightly greater than 0, and zϭ ͑a͒ 0.01, ͑b͒ 0.1, ͑c͒ 0.2, ͑d͒ 0.5, ͑e͒ 1.0.
Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone 3145
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 1 a 0ϭ 2 ͑r,0͒ 2rdr ͑a Ϫ1͒ ͵1
ϱ 4 bnJ1͑n͒ ϭϪ 2 ͚ sinh͑nZ͒, ͑53͒ a Ϫ1nϭ1 n and it is again convenient to work with the velocity potential FIG. 5. An array of discs of radius R, spaced at intervals 2Z along the centreline of a pipe of radius aR. ⌽zϭϪ0 , ͑54͒
with ⌽zϭ0onzϭ0, and ⌽zϭϯ0 on zϭϮZ. Note that the mean velocity over the pipe cross section is not integrable. This serves as a reminder that in practice a separation is likely to occur at the edge of the disc. 2ruz dr ͵0
ϱ a V. A LINEAR ARRAY OF DISCS ϭϪ 4r nbnJ0͑nr͒cosh͓n͑ZϪz͔͒dr ͵0 n͚ϭ1
Suppose we have a regular array of discs, spaced with ϱ ͑non-dimensional͒ separation 2Z ͑figure 5͒. The radial veloc- ϭ4a͚ bnJ1͑na͒cosh͓n͑ZϪz͔͒drϭ0, ͑55͒ ity will be zero at the mid-point zϭZ between two such nϭ1 discs, and hence, instead of ͑7͒, we look for a velocity po- where we have used ͑8͒. Hence, as in Sec. II, we are in a tential with the dimensionless form frame in which the mean axial velocity of the liquid is zero. ϱ The discs have zero volume, and hence the mean axial ve-
͑r,z͒ϭ2 bnJ0͑nr͒sinh͓n͑ZϪz͔͒,0рzрZ. locity of the mixture is also zero. n͚ϭ1 Taking p1 as the pressure at zϭZ, and p2 as the pressure ͑47͒ at zϭϪZ, then evaluating Bernoulli’s equation ͑27͒ at z We must still satisfy the boundary conditions ͑6a–c͒, and ϭϮZ, rϭ0, gives hence ͑9a,b͒ are replaced by ˜ 2 u z p1 p2 ϱ CϪ ϩUu˜ ϭ ϪRU˙ ϭ ϩRU˙ , ͑56͒ 2 z 0 0 Ϫ1ϭ2 ͚ bnnJ0͑nr͒cosh͑nZ͒,0рrϽ1, ͑48a͒ nϭ1 ˜ where u z is the ͑dimensional͒ fluid velocity at rϭ0, ϱ zϭϮZ. Hence, as in ͑33͒, 0ϭ2 ͚ bnnJ1͑nr͒sinh͑nZ͒,1рrϽa. ͑48b͒ nϭ1 p Ϫp ˙ 1 2 RU0ϭ . ͑57͒ In order to ensure that ͑48b͒ is satisfied identically, we take 2 ϱ If the array of discs accelerates, the reaction on each disc is J2mϩ2Ϫp͑n͒ ϭ 2nbn sinh͑nZ͒ cm 1Ϫp 2 . ͑49͒ ϱ m͚ϭ0 1 n J2͑na͒ ˙ 3 Gzϭ2UR ͵ ͚ 2bnJ0͑nr͒sinh͑nZ͒Ϫ0 2rdr Substituting ͑49͒ into ͑48a͒ we find that ͑13͒ is replaced by 0 ͫ nϭ1 ͬ ϱ ϱ ϱ a2 b J ͑ ͒sinh͑ Z͒ cmJ2mϩ2Ϫp͑n͒J0͑nr͒coth͑nZ͒ ˙ 3 n 1 n n Ϫ1ϭ , ϭ8UR 2 ͚ , ͑58͒ ͚ ͚ 1Ϫp 2 ͩa Ϫ1ͪnϭ1 n ϭ ϭ n 1 m 0 n J2͑na͒ and the jump in pressure on rϭ0 between zϭϮZ is 0рrр1. ͑50͒ ˙ ϱ Hence, following the steps outlined in ͑14–19͒, we obtain 8UR bnJ1͑n͒sinh͑nZ͒ p Ϫp ϭ2U˙ R ϭϪ 1 2 0 2 ͚ the linear equations a Ϫ1nϭ1 n ϱ G A c ϭB , jϭ0,1,2, ..., 51 z ͚ jm m j ͑ ͒ ϭϪ 2 2 . ͑59͒ mϭ0 a R where Figure 6 shows the dimensionless acceleration reaction ϱ Gˆ ϭϪG /U˙ R3 as a function of disc separation Z, for the J ͑ ͒J ͑ ͒coth͑ Z͒ z z 2mϩ2Ϫp n 2jϩ1Ϫp n n ˆ A jmϭ Ϫ , ͑52͒ cases aϭ1.1, 1.5, 2.0 and 3.0. Note that Gz decreases as Z ͚ϭ 2 2p 2 n 1 n J2͑na͒ decreases, because the discs shield one another. Figure 7 and the B j are given by ͑23͒, as before. Note that ͑52͒ re- shows similar results for the acceleration reaction per unit ˆ duces to ͑22͒ in the limit Z ϱ. The potential 0 over r length Gz/2Z, proportional to the pressure drop per unit Ͼ1 in the plane of the disc→ is evaluated, as in ͑24͒,bythe length, which increases as Z decreases. Thus the liquid and integral discs become more tightly coupled, per unit length of pipe,
3146 Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp FIG. 8. Computation of drift. The disc ͑a͒ is shown in its initial position z ϭz1; the broken line ͑b͒ shows the initial position of the marked fluid particles at zϭ0; line ͑c͒ shows the final position of the marked particles after the disc has moved off the figure to the right.
ˆ 2 Gz a ϭ . ͑62͒ 2Z a2Ϫ1 The full numerical results agree well with ͑62͒ in the limit ˆ FIG. 6. The non-dimensional added mass coefficient Gz , as a function of Z 0, as indicated in figure 7. the disc separation 2Z. Pipe radius aϭ ͑a͒ 1.1, ͑b͒ 1.5, ͑c͒ 2.0, ͑d͒ 3.0. → VI. DRIFT when either the diameter of the discs increases relative to If a particle of volume V moves along a pipe of cross- that of the pipe, or the number density of discs increases. We sectional area A filled with fluid at rest at infinity, the mean would expect this to be a general feature of suspensions of displacement of the fluid particles will be ϪV/A, as dis- arbitrarily shaped particles. cussed by Eames et al.22 and Benjamin.23 However, the par- When the discs are very close together, we might expect ticle in some sense carries with itself a volume of liquid that the fluid between the discs in rϽ1 will, like the discs, 3 ˆ R Gz corresponding to the added mass of the particle; the have acceleration U˙ , and that the fluid in the annulus rϾ1 background displacement of fluid, away from the wake of ˙ 2 3 ˆ will have a mean acceleration ϪU/(a Ϫ1). We assume that the particle, should therefore be Ϫ(VϩR Gz)/A. This line the disc has no mass, so that the force required to accelerate of argument has been used by Kowe et al.24 to discuss the the disc is equal and opposite to the acceleration reaction motion of bubbles relative to the interstitial fluid, and we Gz . Equating the forces to mass accelerations in a unit cell examine these arguments for the case of a single disc moving ϪZрzрZ, for the central core rϽ1, in a tube. We now change to coordinates fixed in space, rather than 2Z U˙ ϭϪG Ϫ͑p Ϫp ͒, ͑60͒ l z 1 2 fixed in the disc ͑as previously͒. A line of marked fluid par- and for the outer annulus 1ϽrϽa, ticles is placed at zϭ0. The motion of the fluid particles is followed while the disc moves from an initial position z ˙ 1 2ZlU Ͻ0 to a final position z2Ͼ0: a marked fluid particle initially Ϫ ϭϪ͑p1Ϫp2͒. ͑61͒ a2Ϫ1 at (rЈ,0) eventually moves to (r,d) ͑see figure 8͒. Marked particles which are close to the axis of the pipe spend a Hence
FIG. 9. The final positions of marked particles zϭd(r), initially on the ˆ FIG. 7. As for figure 6, showing Gz/2Z as a function of Z. Broken lines plane zϭ0, after motion of a disc from zϭz1ϭϪmax(20,10a)tozϭz2 indicate the limit Z 0 given by ͑62͒. ϭϪz . The pipe radius is ͑a͒ aϭ4.0, ͑b͒ 1.5, ͑c͒ 1.1. → 1
Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone 3147
Downloaded 20 Jun 2003 to 140.247.59.174. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp transport of bubbles by transient large eddies in multiphase turbulent shear flow,’’ in International Conference on the Physical Modelling of Multi- phase Flow, Coventry ͑BHRA Fluid Engineering, Cranfield, England, 1983͒, p. 169. 3A. S. Sangani, D. Z. Zhang, and A. Prosperetti, ‘‘The added mass, Basset, and viscous drag coefficients in non-dilute bubbly liquids undergoing small-amplitude oscillatory motion,’’ Phys. Fluids A 3, 2955 ͑1991͒. 4G. K. Batchelor, An Introduction to Fluid Dynamics ͑Cambridge Univer- sity Press, Cambridge, 1973͒. 5D. Lhuillier, ‘‘Forces d’inertie sur une bulle en expansion se de´plac¸ant dans un fluide,’’ Comptes Rend. Acad. Sci. Paris Se´rII295,95͑1982͒. 6T. R. Auton, J. C. R. Hunt, and M. Prud’homme, ‘‘The force exerted on a body in inviscid unsteady non-uniform rotational flow,’’ J. Fluid Mech. 197, 241 ͑1988͒. 7A. S. Sangani and A. Prosperetti, ‘‘Numerical simulation of the motion of particles at large Reynolds numbers,’’ in Particulate Two-Phase Flow, edited by M. C. Roco ͑Butterworth-Heinemann, Boston, 1993͒, p. 971. 8N. Zuber, ‘‘On the dispersed two-phase flow in the laminar-flow regime,’’ ˆ Chem. Eng. Sci. 19, 897 1964 . FIG. 10. ͑a͒ The non-dimensional added mass coefficient Gz of a single ͑ ͒ 9 disc, as a function of pipe radius a. ͑b͒ The backwards background volume R. Ishii, Y. Umeda, and N. Shishido, ‘‘Bubbly flows through a flux Ϫ2a2d(a) based on the wall drift d(a). converging–diverging nozzle,’’ Phys. Fluids A 5, 1630 ͑1993͒. 10L. van Wijngaarden, ‘‘Hydrodynamic interaction between gas bubbles,’’ J. Fluid Mech. 77,27͑1976͒. 11 considerable time in the vicinity of the stagnation points at W. R. Smythe, ‘‘Flow around a sphere in a circular tube,’’ Phys. Fluids 4, 756 ͑1961͒. the center of the front and rear faces of the disc. The dis- 12W. R. Smythe, ‘‘Flow around a spheroid in a circular tube,’’ Phys. Fluids placement d of such particles is therefore large, with d 7, 633 ͑1964͒. ϳlog(r), as discussed by Lighthill25 for the case of drift 13X. Cai and G. B. Wallis, ‘‘Potential flow around a row of spheres in a around a spherical particle. circular tube,’’ Phys. Fluids A 4, 904 ͑1992͒. 14X. Cai and G. B. Wallis, ‘‘The added mass coefficient for rows and arrays The initial and final positions of the disc were taken to of spheres oscillating along the axes of tubes,’’ Phys. Fluids A 5, 1614 be at zϭϮmax(20,10a), and in figure 9 we show the final ͑1993͒. displacements d(r), for pipes with non-dimensional radius 15J. C. Cooke and C. J. Tranter, ‘‘Dual Fourier–Bessel series,’’ Q. J. Mech. aϭ4, 1.5 and 1.1. The average displacement ͐ad(r)rdr was Appl. Math. 12, 379 ͑1959͒. 0 16I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory computed: a numerical integration under the curves of figure ͑North-Holland, Amsterdam, 1966͒. 9 for rϾ0.06 was cancelled, to within about 3% for aϽ3, by 17W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for an estimate of the integral of the logarithmic singularity in the Special Functions of Mathematical Physics ͑Springer-Verlag, Berlin, 1966͒. the region rϽ0.06. This was a useful check on the accuracy 18 of the integration of the trajectories of marked particles. H. Lamb, Hydrodynamics, 6th ed. ͑Cambridge University Press, Cam- bridge, England, 1932͒. The disc has zero volume V, and hence when the pipe 19J. D. Sherwood and H. A. Stone, ‘‘Electrophoresis of a thin charged disc,’’ radius a is sufficiently large the background drift Phys. Fluids 7, 697 ͑1995͒. Ϫa2d(a) should equal the non-dimensional added mass 20S. K. Lucas and H. A. Stone, ‘‘Evaluating infinite integrals involving ˆ Bessel functions of arbitrary order,’’ J. Comput. Appl. Math. 64, 217 Gz . In figure 10 we see that this is approximately true for ͑1995͒. aϾ5: wall effects become important when aϽ5. 21S. K. Lucas, ‘‘Evaluating infinite integrals involving products of Bessel functions of arbitrary order,’’ J. Comput. Appl. Math. 64, 269 ͑1995͒. 22 ACKNOWLEDGMENT I. Eames, S. E. Belcher, and J. C. R. Hunt, ‘‘Drift, partial drift and Dar- win’s proposition,’’ J. Fluid Mech. 275, 201 ͑1994͒. This collaboration has been supported by the NATO 23T. B. Benjamin, ‘‘Note on added mass and drift,’’ J. Fluid Mech. 169, 251 Collaborative Research Grant Programme ͑CRG.961165͒. ͑1986͒. 24R. Kowe, J. C. R. Hunt, A. Hunt, B. Couet, and L. J. S. Bradbury, ‘‘The effects of bubbles on the volume fluxes and the pressure gradients in 1R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops and Particles unsteady and non-uniform flow of liquids,’’ Int. J. Multiphase Flow 14, ͑Academic Press, New York, 1978͒. 587 ͑1988͒. 2N. H. Thomas, T. R. Auton, K. Sene, and J. C. R. Hunt, ‘‘Entrapment and 25M. J. Lighthill, ‘‘Drift,’’ J. Fluid Mech. 1,31͑1956͒.
3148 Phys. Fluids, Vol. 9, No. 11, November 1997 J. D. Sherwood and H. A. Stone
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