High-Precision Numerical Scheme for Vortical Flow

Transcription

1 Applied Mathematis, 013, 4, Published Online Otober 013 ( High-Preision Numerial Sheme for Vortial Flow Kei Ito 1*, Tomoaki Kunugi, Hiroyuki Ohshima 1 1 Adaned Nulear System Researh and Deelopment Diretorate, Japan Atomi Energy Ageny, Ibaraki, Japan Department of Nulear Engineering, Kyoto Uniersity, Kyoto, Japan * ito.kei@jaea.go.jp Reeied June 10, 013; reised July 10, 013; aepted July 17, 013 Copyright 013 Kei Ito et al. This is an open aess artile distributed under the Creatie Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, proided the original work is properly ited. ABSTRACT In this study, a new high-preision numerial simulation sheme for ortial flows (ortex-based sheme) is proposed. This sheme identifies a ortial flow in eah omputational ell, and then, reonstruts a ortial eloity distribution based on the Burgers ortex model. In addition, a pressure distribution in the iinity of the ortex enter is also reonstruted. The momentum transfer is alulated with the reonstruted eloity and pressure distributions, and therefore, the ortex-based sheme an simulate ortial flows more aurately than the onentional shemes. In fat, as the simulation result of inisid ortex attenuation problem, the ortex-based sheme shows lower simulation error ompared to the onentional disretization shemes. Moreoer, also in the numerial simulation of the quasi-steady ortial flow, the simulation auray of the ortex-based sheme is superior to those of the onentional shemes. Keywords: Computational Fluid Dynamis; Numerial Sheme; Vortex Caitation; Burgers Vortex Theory 1. Introdution Vortex aitation an be obsered in arious engineering senes, e.g. pump sump. Howeer, in usual, the ourrene of the ortex aitation is not faorable beause the ibration of strutural omponents and/or the noise an be indued by the ortex aitation. Moreoer, the aitation bubbles generated at the ortex ore may damage a strutural surfae when they ollapse. Also in sodiumooled fast reators [1], the sub-surfae ortex aitation might be indued by high-speed sution flow into an outlet pipe in the reator essel. Howeer, the onset ondition of the ortex aitation annot be larified easily beause the ortex aitation shows highly ompliated behaiors assoiated with phase hange. Therefore, the authors hae onduted experimental and numerial studies to ealuate the onset ondition of the ortex aitation. As the experimental study, a fundamental experiment is onduted to inestigate the influene of the dynami isosity on the onset ondition [], and some sale model tests are also onduted to hek the dependeny on the loal strutural geometry and the effet of ountermeasure obstales [3]. On the other hand, in the numerial study, a high-preision simulation algorithm for sub-surfae ortex flows is under deelopment, in whih a unstrutured mesh sheme is employed to model aurately * Corresponding author. the loal strutural geometry near a sub-surfae ortex. The high-preision numerial simulation algorithm is deeloped originally for the ealuation of gas entrainment phenomena in sodium-ooled fast reators [4-6]. Currently, it is onfirmed this algorithm an simulate the gas entrainment phenomena aused by free surfae orties, that is, dynami ortex behaiors, i.e. ortex deelopment, moement and attenuation, an be simulated. Therefore, the algorithm is onsidered to be appliable to numerial simulations of the ortex aitation. Howeer, the numerial simulation of the ortex aitation is not easy beause the sub-surfae ortex has highly thin ortex ore ompared to the free surfae ortex. In other words, it is ery diffiult to reprodue the eloity distribution at the iinity of the ore of the sub-surfae ortex. In this ase, a ery fine mesh at the iinity of the ortex ore and/or a high-preision numerial simulation sheme should be employed to simulate the sub-surfae ortex aurately. A ery fine mesh is easy to be onstruted but requires high omputational ost whih may exeeds the limit of usable omputational resoure. Therefore, a highpreision numerial simulation sheme is neessary to simulate the sub-surfae ortex aurately. Here, it should be noted the implementation of well-known high order (more than nd order) shemes, e.g. K-K sheme [7] are diffiult on unstrutured meshes beause of irregular omputational ell arrangement.

2 18 K. ITO ET AL. In this paper, a high-preision simulation sheme for ortial flows on unstrutured meshes is presented. In onentional shemes, the eloity distribution between omputational ells is interpolated with the onstant, linear or higher order funtions based on the disrete alues defined at the ells, and therefore, the existene of a ortial flow is onsidered indiretly through the eloity distribution around the ortex. On the other hand, in our new high-preision sheme, a ortial flow is identified in eah ell and the ortial eloity distribution at the iinity of the ortex enter is reonstruted loally based on the Burgers ortex theory [8]. Namely, the existene of a ortial flow is onsidered diretly in the new high-preision sheme. In this sense, this new sheme is alled the ortex-based sheme in this paper. In addition to the eloity distribution, the pressure distribution at the iinity of the ortex enter is also onsidered based on the Burgers ortex theory to simulate aurately the mehanial balane between pressure gradient and entrifugal fore. Then, the alulations of momentum transport through ell faes are performed in onsideration of the reonstruted eloity and pressure distributions. The superiority of the ortex-based sheme to the onentional shemes is onfirmed by the fundamental erifiations, suh as the numerial simulation of ortex attenuation.. Formulation of High-Preision Numerial Sheme for Vortial Flow (Vortex-Based Sheme).1. The Identifiation of a Vortial Flow in Eah Computational Cell In the ortex-based sheme, the existene of a ortial flow is onsidered diretly. Therefore, in the first plae of the alulation, ortial flows are identified by applying the disriminant [9,10] to eah omputational ell. Namely, a ortex enter exists in a ell when the following disriminant is satisfied in the ell. D Q R , (1) where Q and R are the seond and third inariants of the eloity gradient tensor Aij u i, j defined in eah ell. 3 Q A A, () i, j1 ij ji R det A, (3) In the ell with a ortex enter, the diretion of the ortex, i.e. the diretion of the rotational axis, is obtained as the eigen etor of the eigenalue equation of the eloity gradient tensor. Then, a plane is defined, whih is normal to the diretion of the ortex enter and passes through the ell enter. On this plane, the point with zero in-plane eloity is determined as the ortex enter... The Supplement of the Vortial Veloity Distribution A strong ortial flow has ery large eloity gradient in the immediate iinity of the ortex enter. Therefore, it is highly diffiult to simulate suh a eloity distribution aurately with limited omputer resoures (lak of suffiient number of ells). In the ortex-based sheme, the ortial eloity distribution is supplemented in the iinity of the ortex enter by utilizing the Burgers ortex model whih is known as an exellent model of the orties in nature, e.g. free surfae orties, sub-surfae orties or turbulent orties. The irumferential eloity u distribution of the Burgers ortex model is written as r u r 1exp, (4) r r0 where r is the radial oordinate with the origin at the ortex enter. Equation (4) has two parameters, i.e. the irulation Γ and the speifi radius r 0. In the ortexbased sheme, these parameters are determined to make the irumferential eloity distribution onsistent with the loal eloity distribution. Namely, upon a gien (simulated) eloity field, two parameters are adjusted to minimize the differene in the irumferential eloity distributions of Equation (4) and the gien field at the ells in the iinity of a ortex enter. The alulation proedure is shown in Figure 1 as follows: 1) a ell with a ortex enter (alled ortex ell in this paper) is seleted; ) erties on the ortex ell and the ells around the ortex ell is seleted; 3) when the eloity is gien at the ell enters, the irumferential eloities at the erties u are inter polated from the ell enter eloities u as follows: wu surround u, (5) w surround where the summations are operated on all ells surrounding eah seleted ertex, and w is the weighting fator for the summation; 4) the differenes between the Burgers model (Equation (4)) and the irumferential eloities at the seleted erties u are alulated and aeraged as the form of the weighted root mean square (shown by an oerbar) as follows: u w u erties, (6)

3 K. ITO ET AL. 19 Vortex enter Vortex ell (a) (b) () Burgers u (d) Figure 1. Proedure to determine ortex parameters: (a) seletion of a ortex ell; (b) listing up of erties; () interpolation of irumferential eloity at eah ertex (red etors); (d) alulation of differenes at eah ertex (Burgers ortex eloity in horizontal is shown by blue ure) and (e) adjustments of parameters. (e) where the summation is operated on all seleted erties, and w is the weighting fator for the summation; 5) the irulation and the speifi radius are adjusted iteratiely to minimize the aeraged differenes u. The weighting fator in Equation (5) is determined by minimizing the ost-funtion (Co in Equation (7)) under the onstraint formulated as Equation (8) [11], and the weighting fator in Equation (6) is determined in a similar manner. surround 1 w Co r surround, (7) w r 0 (8) where r is the etor joining a ell enter to a ertex. In Figure 1, the alulation proedure on a two-dimensional strutured mesh is shown for simplifiation. Howeer, the alulation on a three-dimensional unstrutured mesh an be onduted in the same proedure..3. The Momentum Transport Calulation The finite olume method (FVM) is employed frequently in the numerial simulations on unstrutured meshes. In that ase, eah ell is generally seleted as the ontrol olumes for disretization. Also in our high-preision numerial simulation algorithm, suh simulation method (ell based FVM) is employed, and therefore, the following desription is gien based on the ell based FVM. Howeer, the onept of the ortex-based sheme is appliable also to other methods, e.g. the finite differene method. When the ell based FVM is employed on an unstrutured mesh, it is diffiult to formulate high order momentum transport shemes, i.e. third or higher order shemes, due to the irregular ell arrangement. Therefore, the first and seond order upwind shemes are used frequently for the momentum transport alulation. In the first order upwind sheme, the momentum on a ell fae is determined to be same as that in the upwind ell of the ell fae. On the other hand, in the seond order upwind sheme, f the momentum on a ell fae m is alulated in onsideration of the momentum ( m ) and momentum m in the upwind ell of the ell gradient fae: f m m m r f, (9)

4 0 K. ITO ET AL. where r f is the etor joining the upwind ell enter to the ell fae enter. In Equation (9), the momentum gradient an be alulated by the Gauss-Green method [11] or Least-square method [1]. The first or seond order upwind sheme works well in the numerial simulations of smooth flows without disontinuity (shok) and strong orties. Howeer, as mentioned briefly in Setion., the eloity distribution in the iinity of a ortex enter is ery sharp and a ery high resolution mesh is neessary to simulate the ortial eloity distribution aurately by the first or seond order upwind sheme. Instead of suh a ery high resolution mesh, whih is not appliable to pratial simulations, the ortex-based sheme supplements the ortial eloity distribution in the iinity of a ortex enter by the Burgers ortex model (Equation (4)) and alulates the momentum transport based on the supplemented eloity distribution. Namely, when the eloity distribution in the iinity of a ortex enter is gien by Equation (4), the momentum in the irf umferential diretion m is alulated on a ell fae as m u, (10) f f f f f where is the fluid density on the ell fae and u is the irumferential eloity at the ell fae enter alulated by Equation (4). Then, the momentum transport is alulated based on Equation (10)..4. The Pressure Calulation In the iinity of a ortex enter, not only the eloity gradient but also the pressure gradient is large due to the mehanial balane between the irumferential eloity and the pressure gradient (see Equation (11)). Therefore, the pressure should be supplemented to simulate ortial flows aurately. In the ortex-based sheme, the pres- sure distribution in the iinity of a ortex enter is alulated in onsideration of the mehanial balane equation of the irumferential eloity and the pressure gradient, written as u r p, (11) r where p is the pressure. Substituting the irumferential eloity distribution of the Burges model (Equation (4)) into Equation (11) leads the pressure distribution. R 3 1 exp d r, (1) R 0 R p R R R where R r r0. The pressure alulated by Equation (1) is applied to the faes on a ortex ell and the faes on the ells around the ortex ell. Then, the pressure gradient term in the momentum transport equation (Naier-Stokes equation) is alulated at eah ell based on the applied pressure alues. 3. Verifiation 3.1. The Reproduibility of a Vortial Flow As the first erifiation, the reproduibility of a ortex by the ortex-based sheme is addressed. In this alulation, the Burgers ortex is set on two-dimensional strutured meshes, and then, the identifiation and the eloity supplement alulations are onduted to hek the reproduibility of the ortex. As shown in Figure, the simulation domain is and subdiided into 8 8 square ells. The eloity distribution of the Burgers ortex is applied to all ells with arious ortex enter positions, speifi radii and/or irulation alues. Table 1 shows the alulation results. It is eident that the ortex enter position and the irulation are well re- Table 1. Reproduibility of Burgers ortex. Original (gien) alues Calulated alues Vortex enter oordinate Cirulation Speifi radius Vortex enter oordinate Cirulation Speifi radius (0.00, 0.00) (0.00, 0.00) (0.01, 0.00) (0.0094, ) (0.01, 0.01) (0.0103, ) (0.05, 0.05) (0.0503, ) (0.00, 0.00) (0.00, 0.00) (0.00, 0.00)

5 K. ITO ET AL. 1 y x Figure. Simulation mesh and original Burgers ortex with Γ = and r 0 = produed in all alulation ases regardless of the ariations of the ortex enter position, irulation and/or speifi radius of the original Burgers ortex. As for the alulated speifi radius, almost the same alues are ealuated regardless of the ariations of the ortex enter position and/or irulation. Howeer, the alulation auray (reproduibility) is affeted signifiantly by the speifi radius of the original Burgers ortex. In other words, the alulation error is about 5% when the original speifi radius is the same as the ell size (0.15), and the error is redued to about 6.8% when the original speifi radius is twie the ell size. In pratial simulations, this kind of loal (only in the iinity of a ortex) alulation error may be negligible een if it reahes about 5%, beause muh larger error soures an exist in suh simulations. Howeer, the alulation results show that the mesh resolution should be at least half of the speifi radius of a ortex to perform aurate simulations. This onstrain may seem ery diffiult to be satisfied. Howeer, the authors preious researh shows that the ell size has to be about 10 of the speifi radius to reprodue the ortial eloity distribution when the seond order upwind sheme is employed [4]. Therefore, the neessary number of ells for aurate simulations an be redued signifiantly by employing the ortex-based sheme. 3.. The Simulation of Attenuation Behaiors of an Inisid Vortex It is well known that the momentum transport shemes, e.g. the first order upwind sheme, hae the numerial isosity dependent on trunation errors of eah sheme. Therefore, een in the numerial simulations of an inisid ortex, the ortial eloity distribution deays temporally and the total kineti energy in the simulation domain is not onsered. In this sense, the onseratieness of the total kineti energy an be an indiator of the influene of the numerial isosity, that is, the simulation auray. Here, attenuation behaiors of an inisid or- tex are simula ted with four shemes, i.e. the first order upwind, seond order upwind, third order M USCL and ortex-based shemes, and the simulation results are ompared in terms of the onseratieness of the total kineti energy. The numerial simulations are onduted on two-dimensional domain subdiided into square ells (shown in Figure 3). As for the mesh resolution, four strutured meshes, i.e. 8 8, 16 16, 3 3 and square ells, are employed. At the initial state, the eloity distribution of the Burgers ortex with the irulation of and the speifi radius of is applied to the meshes, and then, the simulations of 100 time-marhing with the time inrement of 0.01 are onduted. The temporal ariations of the eloity distributions and the total kineti energies are inestigated in eah simulation for the omparison of four shemes. In Figure 4, the temporal ariations of the eloity distributions on the 3 3 mesh are shown. The simulation result with the first order upwind sheme (Figure 4(a)) shows rapid deay of the initial ortial eloity distribution. When the seond order upwind shemes is employed, suh a deay behaior is suppressed and the peak of the irumferential eloity in the iinity of the ortex enter is maintained appreiably een at t = The simulation result with the third order MUSCL sheme is almost the same as the result with the seond order upwind sheme. The ortex-based sheme shows superior simulation auray to these three shemes. The deay of the irumferential eloity distribution in the iinity of the ortex enter is apparently small ompared to the simulation result with the seond order upwind sheme. The temporal ariations of the total kineti energies on the 3 3 mesh are shown in Figure 5. In the simulation with the first order upwind sheme, the total kineti energy dereases ery rapidly along with the rapid deay of the ortial eloity distribution as shown in Figure 4(a). The seond order upwind sheme highly improes the total energy onseration ompared to the Figure 3. Simulation mesh and initial eloity distribution on 3 3 mesh.

6 K. ITO t = 0.10 ET AL. t = 0.0 t = 0.30 t = 0.40 t = 0.30 t = 0.40 t = 0.30 t = 0.40 (a) t = 0.10 t = 0.0 (b) t = 0.10 t = 0.0 () Figure 4. Temporal ariations of eloity distributions on 3 3 mesh: (a) first order upwind sheme; (b) seond order upwind sheme and () ortex-based sheme. first order upwind sheme. The loss of the total kineti energy at t = 1.0 is about 30% of its original alue (t = 0.0), whih is about 48% in the simulation with the first order upwind sheme. The third order MUSCL sheme shows slightly higher total kineti energy onseration than the seond order upwind sheme, and the loss of the total kineti energy at t = 1.0 is about 6%. Howeer, the temporal ariation of the total kineti energy is similar to that in the simulation results with the seond order upwind sheme. Compared to the other three shemes, the ortex-based sheme shows muh better total energy onseration. In fat, the loss of the total kineti energy at t = 1.0 is about 17%. Therefore, the ortex-based sheme has a superior ability to simulate ortial flows aurately. It should be noted that the pressure supple mentation (Setion.4) is important as well as the eloity supplementation (Setion.3), beause the loss of the total kineti energy at t = 1.0 inreases to about 3% when the pressure supplementation is not employed. In the ortex-based sheme, two parameters, i.e. irulation and speifi radius, dominate the ortial eloity distribution. Here, the temporal behaiors of these two parameters are inestigated as shown in Figure 6. When a relatiely oarse mesh (16 16 mesh) is employed, the speifi radius at the initial state is ealuated as a onsiderably larger alue than that of the original Burgers ortex and the speifi radius beomes larger with time. On the other hand, the irulation is ealuated aurately at the initial state and does not derease signifiantly with time. When a finer mesh (3 3 mesh) is employed, the

7 K. ITO ET AL. 3 Total kineti energy Vortex-based 1st-order upwind nd-order upwind 3rd-order MUSCL Time Figure 5. Simulation mesh and initial eloity distribution on 3 3 mesh. Speifi radius Cirulation Initial alue 16 x 16 mesh 3 x 3 mesh Time (a) Initial alue 16 x 16 mesh 3 x 3 mesh Time (b) Figure 6. Temporal behaiors of ortex parameters: (a) speifi radius and (b) irulation. ealuation results of these two parameters hanges ompletely. In the finer mesh ase, the speifi radius is ealuated aurately throughout the simulation beause the enhaned mesh resolution improes the reproduibility of the ortial flow (as shown in the preious setion). Therefore, the loss of the kineti energy is indued mainly by the hange in the irulation. In fat, the irulation dereases temporally with the deay of the peak irumferential eloity. These ealuation results of the speifi radius and the irulation show the harateristis of the ortex-based sheme an be hanged by mesh resolution. In general, the mesh resolution has a signifiant effet on the simulation auray. Figure 7 shows the loss of the total kineti energy (ratio from its original alue as the funtions of the ell size Δ) in the simulations results on 8 8, 16 16, 3 3 and meshes. The simulation auray is highly enhaned with the mesh resolution when the first order upwind, seond order upwind or third order MUSCL shemes is employed. On the other hand, the ortex-based sheme gies a superior simulation auray of ortial flows on both the oarse and refined meshes. In other words, the ortex-based sheme shows roughly the same simulation auray both on the oarse and fine meshes. Therefore, the ortex-based sheme is espei ally useful in the simulations on rather oarse meshes, on whih other shemes fails to gie aurate simulation results. 4. Numerial Simulation of Quasi-Steady Vortex To hek the appliability of the ortex-based sheme to the numerial simulation on a three-dimensional unstrutured mesh, the quasi-steady ortex is simulated in referene to the experiment by Moriya [13]. Figure 8 shows the simulation mesh, whih onsists of a ylindrial tank with the diameter of 0.40 m, an inlet slit with the width of the 0.04 m and an outlet nozzle with the diameter of 0.05 m. The inlet slit is installed tangentially to the y- Ratio Vortex-based 1st-order upwind nd-order upwind 3rd-order MUSCL r 0 / Figure 7. Loss of total kineti energy. (a) Outlet nozzle Cylindrial tank (b) Inlet slit Figure 8. Simulation mesh of the quasi-steady ortex experiment by Moriya: (a) top iew and (b) bird s eye iew.

8 4 K. ITO ET AL. lindrial tank and the outlet nozzle is installed on the bottom of the ylindrial tank. The uniform flow through the inlet slit generates a ortial flow in the ylindrial tank and the strength of this ortex is enhaned by the downward flow towards the outlet nozzle. The working fluid is water at room temperature. Two ases of unsteady numerial simulations are onduted with the seond order upwind sheme and with the ortex-based sheme until quasi-steady states are obtained. As shown in Figure 8, the mesh resolution is rather low, that is, the horizontal ell size at the enter of the ylindrial tank is about 5.0 mm, whih is not small enough to the measured speifi radius (about 6.0 mm). In this study, a gasliquid interfae loated in the Moriya s experimental apparatus is not onsidered, and only the omparison of the simulation results is onduted. All strutural walls are modeled as non-slip walls. As the other boundary onditions, a uniform inlet eloity with the flow rate of m 3 /s is applied to the inlet a nd a onstant pressure ondition is applied to the outlet. Figure 9 shows the horizontal eloity distributions at the height of 0.15 m from the bottom of the ylindrial tank. It is eident that the or tex-based sheme gies larger eloity in the iinity of the ortex enter ompared to the seond order upwind sheme. The differene in the simulation results with these two shemes an be obsered learly in the omparison of the irumferential eloity distributions (shown in Figure 10). The ortex-based sheme gies larger peak of the irumferential eloity and smaller speifi radius. Namely, the peak of the ortial eloity in the iinity of the ortex enter is simulated more sharply by employing the ortex-based sheme instead of the seond order upwind sheme. Therefore, it is onfirmed that the ortex-based sheme an simulate ortial flows aurately een on three-dimensional unstrutured meshes. 5. Conlusion In this paper, the ortex-based sheme is deeloped to ahiee high-preision numerial simulations of strong ortial flows whih may ause the ortex aitation. In (a) Figure 9. Horizontal eloity distributions: (a) seond order upwind sheme and (b) ortex-based sheme. (b) Cirumferential eloity nd order upwind Vortex-based Radial oordinate Figure 10. Comparison of irumferential eloity distributions. the ortex-based sheme, at the first plae, a ortial flow is identified in eah omputational ell, and then, the Burgers ortex model is applied to supplement a ortial eloity distribution in the iinity of the ortex enter. The pressure distribution in the iinity of the ortex enter is also supplemented. The momentum transfer is alulated with those supplemented eloity and pressure distributions. The simulation result of the inisid ortex attenuation problem shows the ortex-based sheme leads lower si- mulation error ompared to the first to third order shemes. In addition, suh a low simulation error is obtained regardless of the mesh resolution. Therefore, the ortexbased sheme an simulate ortial flows more aurately than the onentional shemes. The higher simulation auray of the ortex-based sheme than the onentional sheme is shown also in the numerial simulation of the quasi-steady ortex experiment. The ortexbased sheme sueeds in simulating the ortial flow on a three-dimensional unstrutured mesh more aurately than the seond order upwind sheme. This result implies that the ortex-based sheme is appliable to pratial simulations. REFERENCES [1] K. Aoto, N. Uto, Y. Sakamoto, T. Ito, M. Toda and S. Kotake, Design Study and R&D Progress on Japan Sodium- Cooled Fast Reator, Journal of Nulear Siene and Tehnology, Vol. 48, No. 4, 011, pp [] T. Ezure, N. Kimura, A. Tobita and H. Kamide, Influenes of Fluid Visosity on the Ourrenes of Caitation Due to Sub-Surfae Vortex, Proeedings of the Seenth Korea-Japan Symposium on Nulear Thermal Hydraulis and Safety, Chunheon, Noember 010, N7P007. [3] T. Ezure, K. Ito, T. Onojima, N. Kimura and H. Kamide, Fundamental Behaior of Vortex Caitation in a 1/ Saled Upper Plenum Model of Sodium Cooled Fast Reator, Proeedings of the 8th Korea-Japan Symposium

9 K. ITO ET AL. 5 on Nulear Thermal Hydraulis and Safety, Beppu, 9-1 Deember 01, N8P1068. [4] K. Ito, T. Kunugi, H. Ohshima and T. Kawamura, Formulations and Validations of a High-Preision Volume-of- Fluid Algorithm on Non-Orthogonal Meshes for Numerial Simulations of Gas Entrainment Phenomena, Journal of Nulear Siene and Tehnology, Vol. 46, No. 4, 009, pp [5] K. Ito, T. Kunugi and H. Ohshima, High-Preision Reonstrution of Gas-Liquid Interfae in PLIC-VOF Framework on Unstrutured Mesh, In: A. Kuzumin, Ed., Computational Fluid Dynamis 010, Springer, Berlin, 011, pp [6] K. Ito, T. Kunugi and H. Ohshima, A High-Preision Unstrutured Adaptie Mesh Tehnique for Gas-Liquid Two-Phase Flows, International Journal for Numerial Methods in Fluids, Vol. 67, No. 11, 011, pp [7] T. Kawamura, H. Takami and K. Kuwahara, Computation of High Reynolds Number Flow around a Cirular Cylinder with Surfae Roughness, Fluid Dynamis Researh, Vol. 1, No., 1986, pp [8] J. M. Burgers, A Mathematial Model Illustrating the Theory of Turbulene, In: R. Mises and T. Karman, Eds., Adane in Applied Mehanis, Aademi Press ING., New York, 198, pp [9] K. Sawada, A Conenient Visualization Method for Identifying Vortex Center, Transation of the Japan Soiety for Aeronautial and Spae Sienes, Vol. 38, No. 10, 1995, pp [10] D. Sujudi and R. Haimes, Identifiation of Swirling Flow in 3-D Vetor Fields, Tehnial Report AIAA , Amerian Institute of Aeronautis and Astronautis, [11] D. Kim and H. Choi, A Seond-Order Time-Aurate Finite Volume Method for Unsteady Inompressible Flow on Hybrid Unstrutured Grids, Journal of Computational Physis, Vol. 16, No., 000, pp [1] T. J. Barth and D. C. Jespersen, The Design and Appliation of Upwind Shemes on Unstrutured Meshes, Tehnial Report AIAA , AIAA 7th Aerospae Sienes Meeting, Reno, 9-1 January [13] S. Moriya, Estimation of Hydrauli Charateristis of Free Surfae Vorties on the Basis on Extension Vortex Theory and Fine Model Test Measurements, Tehnial Reports of Central Researh Institute of Eletri Power Industry U9707, 1998.