Simulation on motion of particles in vortex merging process

Appl. Math. Mech. -Engl. Ed. 31(4), 461- -470 (2010)Applied MathematicsDOI 10.1007/s10483-010-0406-xand MechanicsCShanghai University and Springer- Verlag(English Edition)Berlin Heidelberg 2010Simulation on motion of particles in vortex merging process*Hai-ming HUANG (黄海明)，Xiao-liang XU (徐晓亮)(Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, P. R. China)(Communicated by Zhe-wei ZHOU)Abstract In a two-phase flow, the vortex merging influences both the flow evolution andthe particle motion. With the blobs splitting -and-merging scheme, the vortex merging iscalculated by a corrected core spreading vortex method (CCSVM). The particle motionin the vortex merging process is calculated according to the particle kinetic model. Theresults indicate that the particle traces are spiral lines with the same rotation directionas the spinning vortex. The center of the particle group is in agreement with that of the .merged vortex. The merging time is determined by the circulation and the initial ratio .of the vortex radius and the vortex center distance. Under a certain initial condition,a stretched particle trail is generated, which is determined by the viscosity, the relativeposition between the particles and the vortex, and the asymmetrical circulation of thetwo merging vortices. .Key words vortex method, vortex merging, particle motion, particle trailChinese Library Classification O359.12000 Mathematics Subject Classification 76B471 IntroductionThe merging of vortex pairs is a common phenomenon when the fAuid flows past a bluntbody. In a two-phase flow, the vortex merging inAuences both the fow evolution and the parti-cle motion. Nowadays, various kinds of fuid numerical methods can be adopted in simulatingthe fAlow past a blunt bodyl1- 4. Generally, vortex methodsl2- 3] become more and more eminentbecause of their high-resolution in treating turbulence problems with high Reynolds numbers.In the last decade, many results(2- 1]1 have been achieved with the vortex method. As a de-terministic vortex method, the core spreading vortex method (CSVM) uses a spreading coretechnique in solving the viscosity equation, which is proven to converge to another equationrather than the Navier-Stokes (N-S) equation unless the core size is strictly controlled. Rossi6]presented a splitting and merging algorithm, and Shielsl'7l improved the merging condition andmade the calculation more accurate. Huangl9] introduced a grid technique, which weakenedthe merging conditions in Rossi's merging algorithm. All of the research provide a solid math-ematical background for understanding the accuracy and stability of the vortex method. Onthe particle motion in vortex structures, Huang and Wu3) presented simulations on the parti-cle motion and distribution for the two-phase flow past a cylinder. Zhang et al.4| studied the* Received Ang. 6, 2009 / Revised Mar. 3, 2010Project supported by the National Natural Science Foundation of China (No. 10572020)Corresponding author Hai-ming HUANG, Ph. D., E-mail: huanghaiming@ tsinghua.org.cn中国煤化工MYHCNM HG462Hai-ming HUANG and Xiao- liang XUparticle motion in a two- phase mixing layer. With the development of the vortex method, thecharacterization on the vortex structure can be revealed with the increasing precision, and moregeneral regularities on the particle motion can be obtained in the vortex merging process, whichis hardly seen in literatures.Recently, with the developments in the aerospace technology, the calculation on a two-phasefow has been increasingly emphasized. Take the denudation in a spacecraft as an example.When a hypersonic vehicle flies, it may be fAushed, and the denudated particles constitute thetwo-phase wake flow with air. Therefore, the particle motion will infuence the distribution onthe ablation and denudation rates. A prerequisite of understanding the denudation mechanismis the calculation and analysis of the particle trajectories. In order to find out the regularity onthe particle motion in a merging process, a corrected core spreading vortex method (CCSVM)with the blobs-splitting- and-merging scheme and the particle kinetic model is used to analyzethe particle motion in the merging process. Finally, we can find some interesting conclusionsthat can provide some references on the safety of spacecrafts.2 Vortex methods in 2D flows2.1 The corrected core spreading vortex methodThe 2D incompressible fow is governed by the vorticity transport equation, which can beexpressed asow)w=vV2w,(1Ot)x8yV2ψ= -w, .(2)where Ux and Uy denote the velocity components on the x and y directions, respectively. 亚is the stream function. Ux = and Uy=-o. v is the kinematic viscosity, and w is thevorticity of the fuid.A Gauss distribution function is introduced. Then, the vorticity field can be dispersed intoN vortex blobs,|x- xil|(3二πσjwhere T; is the circulation of the jth blob, σ; denotes the diameter of the jth blob, x is anarbitrary point in the flow field, and xj denotes the center of the jth blob. From the Biot-Sarvartlaw, the flow velocity in the rectangular coordinate can be calculated as一F，. (y-yi)[1-exp(-- x)，台π? |x- xillo?uy=长r(x-)[-xp(_-..台πx-x;lFj, σj, and axj satisfy Eqs. (4)- (6)5dTj_=0,(4)dσ;= 4v,(5)dt中国煤化工MHCNM HGSimulation on motion of particles in vortex merging process463=u(xj,t).(6It can be seen from Eq. (5) that the characteristic radius σ increases with time. Greengard11jproved that the oversize vortex blobs can lead the solution to a wrong equation rather than theN-S equations. Thus, an exact numerical solution cannot be obtained without a strict control2.2 The blobs-splitting- and-merging schemeOn the basis of Shiels's[7] merging scheme, Huang[9] proposed a splitting and mergingscheme, in which the conservation condition of second -order vorticity moments is fulfilled, anyoversize vortex blob is replaced by M sub-blobs after the control radius σmax is selected, andeach sub-blob has a distance rrad away from the parent blob. The sub-blobs have the sameradius as the parent blob after the splitting process. Let σo and To be the characteristic ra-dius and circulation of the parent blob after the splitting process, respectively. Fe denotes thesub-blob circulation.With the definitionσ0 = aTpar,(7where the subscript par denotes the parent vortex blob, both the coefficients a and M determinea simulation precisionl9l. Larger a and M lead to smaller error. As mentioned above, the secondmoments of vorticity are conserved in a splitting process,=2(1-a2),(8)OparTparTo=(9)Tc == 2MApparently, the amount of vortex blobs increases with an exponential velocity M, which isunsuitable in the numerical simulation. Thus, a merging scheme muust be introduced. Adjacentblobs with the small circulation will be replaced by a new one with the merging conditionl12]defined as下<πε itn时(11)(12)where Te denotes the reference circulation, and E is the governing error.Generally, the zeroth, first, and second moments of vorticity are conserved, and the dis-cretization scheme can be described as(13)j=1MToxo= >Fjxj,(14)j=To_ I;exp|xj-ol\σ中国煤化工MHCNM HG464Hai-ming HUANG and Xiao-liang XU3 The single- particle kinetic modelWithout any consideration of the particle effects on the air fow, the equilibrium equation[13]of the particle in the air flow can be written as1,“=mDt-ma(v,-U(Y,t)- oa2v2U(Y,t). 6πaμX(t)- 6πa2μ“ dX()/dr dr(16)Jo √πv(t- T)with X(t)= V(t)- U(Y(t),t)- ta2V2U(Y,t). Here, V is the gradient operator. mp denotesthe mass of the particle, and mf is the air micelle mass with the same volume of the particle(where the subscript p denotes the particles and f denotes the air). V denotes the particlevelocity. The air velocity U(Y , t) can be obtained by the CCSVM, where Y denotes the positionof the air micelle. μ is the dynamic viscosity. a is the particle radius.品=品+U . V denotesthe material derivative of the air flow, and出=品+ Vp. V denotes the material derivative ofthe particle. -meH(Vp- U(Y ,t)- +a2V2U(Y ,t)) is the buoyancy term. - 6πaμX(t) is theStokes forces. - 6πa2μ fo /dτ denotes the Basset forces. Give the assumption that theparticle radius is small, the minimal terms can be neglected, and Eq. (16) can be simplified as(mp + "n)dV[() = 6ua(U(),t)- V()+ m;U(Y(),t). VU(Y(),t)... 0U(Y(t),t)-mVp(t). VU(Y(t),t)+ meDt(16')Using the dimensionless variables defined by Y - + YL, Vp→VpUo, andt→长，where Land Uo are the typical length and the velocity scales of the air fAow, respectively, we get"W=AIU - W()+B(U+气1(1) Vσ+咒(17)'at6πaμLwith A=(mp+ ImqandB= mp+tmeIn the case of 2D divergence-free Eulerian velocity fields, the motion equation for a passiveparticle isdY(t)= Vp(t).(18)dtAs described in Section 2, with the blobs -splitting and-merging scheme of Eqs. (7)-(15)，thevelocity of the air flow can be acquired through Eqs. (1)-(6). Substituting it into Eq. (17), weget the velocity of the passive particle and the position of the particle motion from Eq. (18)4 Particle trajectories in the blobs-splitting and-merging scheme4.1 Approximation of the initial vorticity fieldA Gauss distribution function is selected to approximate the vorticity field. The sheet-like vortex is dispersed into many vortex blobs impartially, and a stepladder-like distributedvorticity field can be constructed in the zone (Fig. 1). Then, an approximate vorticity field isobtained by assembling some stepladder-like vorticity fields. Suppose that the diffusion obeysdimensionless radius of r = 1.0 as an example. With an initial time to = 0.125, the Oseenvortex is dispersed into 6 682 blobs. As indicated in Fig. 2, the initial vorticity field can beexactly described by the discrete vortex blobs.中国煤化工MHCNM HGSimulation on motion of particles in vortex merging process46525|10.... Approximate valueTheoretical value20F0t100150F93 -2-10.5 1.C2.02.5 3.0Fig.1 Stepladder-like vorticity fieldFig. 2 Precision of the vortex method in sim-ulation of the Oseen vortex4.2 Particle trajectories in the symmetry vortex merging processThe computing field is assumed to be an infinite field. The simulation is performed forPp/Pr= 2.0 with the initial condition (r/b)o= 0.31, To= 200π, and Ot = 0.001 s, wherer is .the vortex radius and b is the distance between the two vortex centers. When 1 681 particlesare evenly distributed in a square domain with a = 0.01 mm, and the collisions among theparticles are not considered, the particle motion can be shown in Fig. 3.6r1= 0.00t= 0.066「t= 0.096-4-20246-6-4-20246-6-4-20246(a)t= 0.00(b)t= 0.06(c)1= 0.09「= 0.18=0.300「t= 0.60 .5-二CS-6-4-2024 6)246-10-)510(d)1= 0.18(e)t= 0.30()i= 0.60.10「1= 0.901= 1.50t= 2.10(C)》入0GD-1010 -10(g)t= 0.90(h)t= 1.50(i)t=2.10Fig. 3 Particle distribution in the merging process with the initial condition (r/b)o= 0.31中国煤化工MYHCNM HG466Hai-ming HUANG and Xiao liang XUAs indicated in Fig. 3, the two vortices distort and rotate around each other as the merg-ing starts. Simultaneously, the particles move to the periphery of the vortex pair (Fig. 3(a)-Fig. 3(b)); then, the vortex radius increases as a result of viscosity (Fig. 3(c)- Fig. 3(f)). In thefinal stage, the merged vortex keeps spinning around its vortex center, and its radius shrinksto a stable value (Fig. 3(g)- Fig. 3()).Compare the results with those computed under different initial conditions of (r/b)o = 0.2ind (r/b)o = 0.4. As indicated in Fig.4 and Fig. 5, with the same circulation condition,the merging process takes the shorter time when the vortex center distance is smaller. Theparticles accumulate at the periphery of the vortex pair and distribute as the loops. A stretchedtrail is formed with a certain initial ratio of the vortex radius and the vortex center distance((r/b)o = 0.31).6pt= 0.00t= 0.08t= 0.2042--6--4-20246-6-4-20246-64-20246(a)i= 0.00(b)t= 0.08(c)i= 0.200pt=0.40 10t=1.00 10pt= 2.005->0-10--5 0510-10--5 0 510 -10)510(d)i= 0.40(e)i= 1.00.(Di= 2.00Fig. 4 Particle distribution in the merging process with the initial condition (r/b)o= 0.2> 0-◎0OO二0(a)1= 0.00(b)1= 0.04()1= 0.085[1=0.14=0.20 10= 1.40CGD一5-10---50510-10一-5一051o(d)1= 0.14(e)1= 0.20(θ1= 1.40Fig.5 Particle distribution in the merging process with the initial condition (r/b)o= 0.4中国煤化工MHCNM HGSimulation on motion of particles in vortex merging process4671.3 Particle trajectories in the asymmetrical vortex merging processNot all the vortices have the same circulation. A much more general case is an asymmetricalvortex merging. Here, we take a sheet-like vortex pair with different circulations of 100π and200π as an example. By the solution procedure mentioned in Section 2, the particle trajectoriesin the merging process can be calculated and displayed in Fig. 6. The vortex with the smallercirculation distorts and moves toward the bigger one, whereas, before the particles reach thestable states, the merging time is smaller than that in the larger circulation case in Fig. 3 andFig. 5. A stretched trail is also generated in this asymmetrical case.5t=0.061=0.10入0O(a) t=0.00(b) t=0.06()t=0.10 .=0.14s「t=0.20心2一(二00(d)1=0.14(e)1=0.18()1=0.20101=0.40t=0.6010p1=1.00 .;上入110一(g)1=0.40(h)1=0.60(i)1= 1.00Fig.6 The particle distribution in the merging process with the initial condition (r/b)o = 0.314.4 The physical mechanism of trailAs indicated in Fig. 3 and Fig. 6, the trails are generated in the late merging stage, whereasno trail is generated in Fig. 4 and Fig. 5. Thus, we can speculate that the physical mechanismof the trails can have a close relationship with the viscosity, the relative position between theparticles and the vortex, and the asymmetrical circulation..4.1 The efect of the viscosity cofficientThe effects of the viscosity cofficient can be analyzed by the calculation of the mergingprocesses in the viscous flow and the inviscid flow, separately. In the inviscid flow, as mentionedin Ref. [14], if the initial condition of (r/b)o ≥(r/b)er is satisfed, the two vortices will merge;otherwise, the two vortices will rotate around each other, where (r/b)er is varied from 0.29 to0.33. In the viscid fow, the vortex radius will increase with time as a result of the viscous中国煤化工MYHCNM HG468Hai-ming HUANG and Xiao-liang XUdiffusion. Thus, the two vortices will merge inevitably after a while. Here, we present thecomputing results in an inviscid flow when the other conditions are the same as those in Fig. 3.As indicated in Fig. 7, no matter what (r/b)o is, the two vortices will not merge in the inviscidThe reason can be found in Eq. (5). The radius of a vortex blob will be a constant with thelapse of time as v = 0. Put σ into Eq. (3). Then, w(x,t) will be unchanged correspondingly.As shown in Fig. 7, no merging process is performed in an inviscid flow. Comparing Fig. 3and Fig. 7, we ensure that the viscosity is a key factor to the trail generation when the otherconditions are maintained.1= 0.00.t= 0.06t= 0.094D(B)>0-6→4→2δ2466←-420246-6-4-20246(a)1= 0.00(b)1= 0.06(C)t= 0.09t=0.30= 0.900「. t= 1.50●●0❷)5t10-50510-10-50510-10一-50510(d)1= 0.30(e)1= 0.90(I1= 1.50Fig.7 Motion of vortex and particles in the inviscid flow (r/b)o = 0.314.4.2 The effect of the relative position between particles and vortexFrom the comparison of Fig.3 and Fig.5 with the same conditions of the circulation and theinitial particle distribution, it can be seen that the difference of the values of (r/b)o is anotherfactor to the particle trails. Calculate the particle traces in a circle particle distribution (Fig. 8)and compare the results with those in Fig. 3. We conclude that the trails in Fig. 3 are resultedfrom the particles in the square corners when the conditions of the circulation and (r/b)o aremaintained."hus, we know that, with the same circulation and the given particle distribution,the difference of the values of (r/b)o is the reason of the generation of trails, whereas the initialparticle distribution is a key factor to the trail when (r/b)o is maintained. In conclusion, therelative position between the particles and the vortex is another reason of the generation oftrails.t= 0.00t= 0.305-一0二06-4-20246-1050510(e)1= 0.30()1= 1.90Fig.8 The particle motion with the initial circle distribution (r/b)o= 0.31中国煤化工YHCNM HGSimulation on motion of particles in vortex merging process4694.5 Particle tracesSelect 20 particles randomly. The radius and the center of the particle group are calculatedwith the least square method (Fig. 9) when the particle motion reaches the stable states. Asindicated in Fig. 10 (only 8 trajectories are displayed in the figure), the trajectory of a singleparticle is a helix.5-xo=-1.052,yo=-0.018,泊-9R=9.4483一05下-10-5-15Fig.9 The least square ftting of the centerFig.10 Traces of the selected particlesand the radius for the particle groupTo find the regularity between the merging and the particle motion, we analyze the particlemotion under four different conditions of the initial asymmetrical circulation. The results areshown in Table 1. Let 5a =∞0-01 describe the precision of the calculation, where xo denotesthe theoretical coordinate of the merged vortex center under conservation conditions of thezeroth and first vorticity momentums, x1 denotes the center of the fitted merged particle group(Fig. 10), R is the fitted radius of the particle group. Compare the theoretical coordinateswith the fitted values in Table 1. It can be concluded that the center of the merged vortex isconsistent with that of the particles group.Table 1Least square fittingof center and radius 0of particlegroup under different merging processesVorticityRζ∞= o-.山l .200π/200π(0.000, 0.000)(0.047, 0.073)8.440 80.088 54%200π/100π(-0.518, 0.045)7.208 20.033 29%500π/100π(- 1.075, 0.000)(-1.052, 0.018)9.44830.009 94%700π/ 100π(-1.210, 0.000)(-1.203, 0.049)10.545 10.023 71%5 ConclusionsOn the basis of the CCSVM and the particle kinetic model, the calculation of the 2D Oseenvortex merging process under symmetry and asymmetry conditions has been performed. Theconclusions can be drawn as follows:(i) The particle trajectories in a merging process are the spiral lines with the same rotationdirection of the spinning vortex.(i) The merging time is determined by the circulation and the initial ratio between thevortex radius and the distance of the vortex pair center. The larger circulation and the largervalue of (r/b)o correspond of to shorter merging time and shorter time before the particlemotion becomes stable.中国煤化工MHCNM HG470Hai-ming HUANG and Xiao-liang XU(ii) A stretched trail is generated in the merging process under certain conditions, whichhas a close relationship with the viscosity cofficient, the relative position between the particlesand the vortex, and the asymmetrical circulation.(iv) When the conservation conditions of the zeroth, first, and second vorticity moments aresatisfied, the center of the particle group is judged as that of the merged vortex.The calculations are carried through the given density ratio and the constant particle size.References[1] Li, Z. H. and Zhang, H. X. Study on gas kinetic algorithm for flows from rarefied transition tocontinuum using Boltzmann model equation. Acta Mechanica Sinica 34(2), 145- 155 (2002)[2] Tong, B. G. and Yin, X. Y. Discussion on vortex methods. 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