THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT.

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1 ISTP-6, 5, PRAGUE 6 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT. Mei-Jiau Huang Mechanical Engineering Department, National Taiwan University mhuang@ntu.edu.tw Tex: fax: Keywords: Vortex merger, Vortex method, Vortex sheet Abstract The merging process of two identical twodimensional vortices is simulated in use of a core-spreading vortex method, which is corrected by employing an element splitting technique to execute regridding and an element combining technique to control the computational amount. Unlike previous investigations that mostly seek for a itical ratio of the characteristic radius to the separation distance of the two vortices in merger for the onset of merging, the present study explores the Lagrangian evolution of each of the vortices and based on the observation proposes a new viewpoint about the merging mechanism. Addressed are the mutual attraction of circulation and the formation of sheet-like vorticity structures. Moreover, viscosity plays the role of making always and faster the attraction of the circulation. Introduction The merging of two vortices into a single one has been observed in a variety of fluid flows. Previous investigations (e.g. Saffman and Szeto, 98 []; Overman nd Zabusky, 98 []; Dritschel, 985 [3]; Griffiths and Hopfinger, 987 [4]; Mitchell and Driscoll, 996 [5]; Meunier and Leweke, [6]; Meunier et al, [7]; Cerretelli and Williamson, 3 [8] and so on) examined the way in which two vortices merge together and determined the itical separation distance for an onset of merger. Basically, the merging process is recognized as follows. At the beginning, two vortices rotate around each other with approximately a constant separation distance. Later, two vortex centers get closer and rapidly merge into a single core. Spiral arms are formed in the outer region at the same time. Eventually, the final vortex is axisymmetrized through filamentation [8, 9]. Although merger has been observed numerically and experimentally, its physical mechanism however is not so clear. Generally, it was believed that when vorticity is advected out of the vortex cores and into the spiral arms, by conservation of angular momentum, the cores correspondingly must move toward each other. Mathematical models have also been developed to predict the merger [7,]. Although the flow structures were analyzed in details, the cause of merger is not ensured either mathematically or physically. Using the vortexin-cell method and taking advantage of the finite-time unstable and stable manifolds associated with transient fixed points, Velasco Fuentes [] dealt with the advection of fluid particles in merger and attempted to quantify the efficiency of merger. His study showed that the formation of spiral arms is not the cause of merger but one of its effects, and that a use of the Lagrangian flow geometry in studying merger is encouraging. In the present work, attempt is to explore the merging mechanism through a Lagrangian study of the flow structures in problem. The disete core-spreading vortex method [] is employed. To control the core widths of computational elements, an element splitting

2 technique, similar to but more accurate than that of Rossi [3], is developed. And to maintain the total number of computational elements reasonably many, an element combining technique, a simplified version of that by Rossi [4], is also employed. The algorithm of selecting elements to be combined is also redesigned for efficiency. This paper is arranged as follows. The numerical methods employed are introduced in Sec.. The Eulerian as well as Lagrangian flow structures of the simulated flow are desibed in Sec.3. A possible cause of vortex merger is claimed based on the flow observations and examined. Conclusions are given in Sec.4. Numerical Methods. The Core-Spreading Vortex Method In the disete vortex method, the vorticity field is composed by N computational elements, which herein have Gaussian spatial distributions, and is written as N x x ω ( xt, ) = exp () πσ σ where x, σ, and = are the center, core width, and strength of the th computational element, respectively. In the core-spreading vortex method, an element moves with the fluid velocity at its center as a whole (that implies deformation is ignored) and its core width expands in time in order to simulate diffusion. In the present study, Leonard s core spreading method [] is employed. That is, the strength of each computational element remains unchanged in time and the core width expands according to d σ = 4 ν () dt where ν is the fluid viscosity. The modified Euler method is then employed to march in time. Noticed is that Leonard s core spreading scheme has been found to have a numerical error on the order of σ, that grows in time according to Eqn.(). To fix this error, some numerical technique is required to keep the core Fig. The example taking M=4 shows how the vortex elements are placed in a splitting event. width small. In the present study, Rossi s element splitting idea [3] is adopted.. Element Splitting Technique To obtain a correct simulation result, a computational element is udged too fat and needs to split into several thinner one when its core width has grown greater than some threshold σ. Let σ be the too fat core width. It is now forced to split into +M elements of width ασ, where α is a free parameter and <α<. One of these thinner elements will replace the original fat element and the others sit around uniformly at a distance r away as illustrated in Fig.. The strength of the central element, the strength of the other elements, and the distance r can then be determined through conservations of the zeroth, nd, and 4 th moments of vorticity, where the n th moment is defined as n x ω dx (3) The results are ( ) = (4) = M (5) ( ) r σ = α (6) where is the circulation of the original fat element. With the extra degree of freedom, the error arising from the splitting process can be made smaller than that associated with Rossi s original scheme.

3 THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT. Attention must be paid to the selected time inement dt in the time-marching scheme. To be consistent, it must be at least smaller than the time interval between two successive splitting processes, that is, approximately ( ) dt <σ α 4 ν (7) In addition, it must be smaller than the time for the first vortex splitting, i.e. ( i ) dt < σ σ 4ν (8) where σ i is the maximum initial core width among the computational elements..3 Element Combining Technique As the time ineases, the total number of the computational elements grows rapidly due to the splitting and blows up eventually. To solve this problem, Rossi [4] suggested to combine similar and close-by elements into one. Let, x, σ be the resulting vortex, where { } = x = x (9) = = ( x x ) σ = σ + = () () Equations (9) to () basically preserve the zero th, st, and nd moments of vorticity. The summation is done over the selected similar and close-by elements. Rossi was next able to show that the so-induced combining error is no greater than M πσ, where M is the maximum absolute value of the following function exp x βexp β( x a) () ( ) ( ) in the parameter domain of b β=σ σ b and a = x x σ R. The disadvantage of Ross s combining method is it poses too many iteria in combining elements. Modifications are made hereafter. First, one notices that M is equal to or β for an arbitrary a. Equation () tells σ >σ for some if all have the same sign. In other words β> and therefore M = β= Maxσ σ (3) for arbitrary a. The combining error becomes thus no greater than πminσ. Furthermore, due to the splitting method used, σ ασ for all as long as only computational elements of the same rotation direction are combined. Consequently, the combining error is no greater than πα σ. If ε is the error tolerance, then the selected similar and close-by computational elements can be combined into one if the following inequalities are fulfilled: <επα σ (4) σ <σ (5) Finally, in order to select the similar and closeby computational elements, the flow domain occupied by all the computational elements will be divided into square cells as shown in Fig.. Computational elements of a same rotation direction will be udged as similar and those within a same cell will be defined as close-by. Experiences show that the combining efficiency is best when the cell size is.5 σ. The possible maximum distance between any pair of closeby computational elements is therefore σ. The procedures of combining Fig. Vortex elements within a same cell are defined as close-by. 3

4 computational elements are summarized as below. (i) Identify computational elements in each cell. (ii) Search for a subset of the computational elements in a cell which has a total circulation close to but less than επα σ. (iii) Compute σ and x. (iv) Replace the whole subset by the one with, σ,x if σ <σ. Go back to (ii), { } otherwise. The iteration of searching for the subset of computational elements may be stopped after several tries; combining is given up if no such a proper subset can be found then. Moreover, in order not to induce too much combining errors, at most one replacement is taken in each cell. 3 Flow Simulation 3. Simulation Conditions The flow simulated herein is the merger of two Burgers vortices. Each of them has a core width of one and a circulation tot. And each is constructed initially by placing computational elements uniformly within a circular region of radius 3. The ratio of their initial core width to the inter-element distance is chosen to be.. Their strengths are then determined by enforcing exact vorticity values at their locations. Parameter values employed in the present simulation are α=.85, M=4, σ =.5, and ε tot =.5%. To obtain the Lagrangian evolution of each Burgers vortex, only one Burgers vortex is calculated and stored in the computer memory. Effects of the other Burgers vortex on the first one are taken into account by the image method. The whole flow is thus made exactly symmetric with respect to the origin. The calculated Burgers vortex is initially located at ( b, ). The initial separation distance between the two Burgers vortex is thus. The fluid viscosity is ν=. b 3. Eulerian Flow Structure Earlier investigations [-] found that merger occurs when the ratio of the core width to the separation distance is about.6~.3. The core width however is not easy to be defined due to the significant shape deformation during evolution. In other words, the effect of shape deformation probably was underestimated in the past. Figure 3 shows the Eulerian evolution of the whole flow structure in which the important ratio is.5 initially and the Reynolds number = πν =6. Note the x-axis is is Re tot chosen to rotate with the line-of-center. As before, two Burgers vortices rotate around each other with a constant separation distance for a while; later, two vortex centers get closer and rapidly merge into a single core with spiral arms formed in the outer region; and the merged vortex gets axisymmetrized gradually through filamentation. Figure 4 measures the time evolutions of the distance between two Burgers b(t), the vorticity value at the origin ω, the maximum vorticity value of the flow ω max, and the angular velocity difference Ω = Ω Ω, where Ω and Ω are two itical angular velocities defined as with S Ω = ω ± (6), S υ u u x y x (7) Note Ω is negative initially and turns to be positive and gradually becomes equal to Ω as two vortices merge. It is known that the flow pattern changes in a rotating reference frame. It can be shown that the origin turns to be a saddle point if the frame rotates at an angular speed Ω in between the above two itical angular velocities; it is a center otherwise. Different flow patterns can be further distinguished when the origin is a saddle, for example with or without an exchange band of vorticity []. For instance, shown in Fig.5 are the flow patterns at t= under several different rotating frames. The two associated itical angular velocities are 3. and 3.6. It is also found that an exchange band can be always found during merging as 4

5 THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT. t = t =.5 t =. t =.4 Fig.3 The evolution of the contours ( ω min = and ω= ) of total vorticity field of the flow with an initial separation b =8 and Re=6. long as a right angular velocity Ω is chosen. Its existence becomes not so itical then. Instead, the angular velocity difference evolves in a way much more correlated to the merging process. Also observed in Fig.4 is that the Burgers vortex centers stop moving toward each other at t. ; thereafter the distance between them remains approximately constant with deeasing oscillations. The final separation is about.8b. Note that the maximum vorticity value does not occur at the origin until t.4. The time period from t. to t.4 was identified as the second diffusive stage by Cerretelli and Williamson [8], during which the location of the maximum vorticity gradually moves from the vortex centers to the origin by diffusion. Fig.4 The time evolutions of the distance between two Burgers ( bb : solid line), the vorticity value at the origin ( ω Re: dotted line), the maximum vorticity value of the flow ω Re: symbols), and the angular velocity ( max difference ( Ω Re: dash line). 5

6 Ω = Ω = 7 Ω = 8 Ω = Ω = 8 Fig. 5 Different flow (streamline) patterns at t= under different rotating frames ( Ω = 3.6 and Ω = 3. ). The gray region in the plot as Ω=7 is the so-called exchange band of vorticity. 3.3 Evolution of Single Vortex The evolution of vorticity contours contributed by a single Burgers vortex is shown in Fig.6. Interestingly, it is seen that the outer contours are attracted by the other Burgers vortex (t=.5) and gradually form a sheet-like structure (t=.8 and t=) besides the center of the other Burgers vortex. The attraction of circulation is attributed to the shape deformation of each vortex. The sheet-like structure then causes the nearby center moving to the right. Conectured is made that the formation of the sheet-like structures is the cause of merging. A detailed calculation of the velocity at the vortex center does show that more than 9 percentages of the merging velocity at all times is contributed by the sheetlike structure. Moreover, the two arms appearing in Fig.6 are obviously of different thickness, implying interestingly again that each spiral arm observed in Fig.3 is contributed mostly from the farther vortex and because of it, the core structure is fatter in one side and thinner in the other. This asymmetry corresponds to the sheet-like structure discussed above. 4. Conclusion In the present study, a purely Lagrangian corespreading vortex method is employed to simulate the merging process of two identical Burgers vortices. Reasonably accurate solutions are attained through a proper splitting technique and a combining method. Flow structures contributed by both Burgers vortices or by ust a single one are explored. The later reveals a new viewpoint about the cause of vortex merger: the mutual attraction of circulations and the formation of sheet-like structures. 6

7 THE CAUSE OF A SYMMETRIC VORTEX MERGER: A NEW VIEWPOINT. t =.5 t =.8 t = t =.4 Fig. 6 The evolution of vorticity contours ( ω min =.5 and ω= 5 ) contributed by a single Burgers vortex. References [] Saffman P.G. and Szeto R., Equilibrium shapes of a pair of equal uniform vortices, Phys. Fluids 3, pp , 98. [] Overman E. A. and Zabusky N.J., Evolution and merger of isolated vortex structures, Phys. Fluid. 5, pp.97-35, 98. [3] Dritschel D.G., The stability and energetics of corotating uniform vortices, J. Fluid Mech. 57, pp.95-34, 985. [4] Griffiths and Hopfinger E.J., Coalescing of geostrophic vortices, J. Fluid Mech. 78, pp.73-97, 987. [5] Mitchell T.B. and Driscoll C.F., Electron vortex orbits and merger. Phys. Fluids 8, 88, 996. [6] Meunier P. and Leweke T., Three-dimensional instability during vortex merging, Phys. Fluids 3, pp ,. [7] Meunier P., Ehrenstein U., Leweke T. and Rossi M., A merging iterion for two-dimensional co-rotating vortices, Phys. Fluids 4, pp ,. [8] Cerretelli C. and Williamson C.H.K., The physical mechanism for vortex merging, J. Fluid Mech. 475, pp.4-77, 3. [9] Melander M.V., McWilliams J.C., and Zabusky N.J., Axistymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation, J. Fluid Mech. 78, pp.37-59, 987. [] Melander M.V., Zabusky N.J., and McWilliams J.C., Symmetric vortex merger in two dimensions: causes and conditions, J. Fluid Mech. 95, pp.33-34, 988. [] Velasco Fuentes O.U., Chaotic advection by two interacting finite-area vortices, Phys. Fluids 3, 9,. [] Leonard A., Vortex methods for flow simulations, J. Comput. Phys. 37, pp , 98. [3] Rossi L.F., Resurrecting core spreading vortex methods: a new scheme that is both deterministic and convergent, SIAM J. Sci. Comput. 7, pp , 996. [4] Rossi L.F., Merging computational elements in vortex simulations, SIAM J. Sci. Comput. 8, pp.4-7,