Shanghai, China; September -6, Effets of the swirl ratio on the turbulent flow fields of tornadolike vorties by using LES turbulent model Zhenqing Liu a, Takeshi Ishihara b a Department of Civil Engineering, Shool of Engineering, The University of Tokyo, 7-3-, Hongo, Bunkyo-ku, Tokyo, Japan b Department of Civil Engineering, Shool of Engineering, The University of Tokyo, 7-3-, Hongo, Bunkyo-ku, Tokyo, Japan ABSTRACT: Tornado-like vorties have been investigated by using LES turbulene model. The flow fields are visualized by virtual water vapor injeted from the ground of the numerial model and the evolution from a single-elled vortex into a multi-vortex onfiguration is suessfully reprodued. The flow fields as well as the fore balane of four typial tornado onfigurations, weak vortex, vortex breakdown, vortex touh-down and multi-vortex are investigated. The definitions for the swirl ratio are summarized and the loal orner swirl ratio is found to be robust and proposed to universalize the researhes. KEYWORDS: Tornado-like vortex, Swirl ratio, Flow fields, LES, CFD simulation. INTRODUCTION Tornadoes are one of the severe natural phenomenons and onsidered as the most violent storm on earth, whih makes it important to take proper onsideration of tornado-indued wind loads and tornado-borne missiles for wind resistant design of strutures. Therefore, detailed information of the three-dimensional flow fields is neessary. Many researhers are motivated to observe the wind dynamis and ollet the data in real tornadoes by using veloity and pressure instrumentation. However, due to the extreme danger faed by the observers, data olletion for tornado s internal flow fields is still not muh of a suess. Reproduing tornadoes experimentally or numerially is therefore an alternative. Laboratory simulations provide a safe, reproduible and ontrollable approah for the tornadorelated researhes. Mitsuta and Monji (984) modified the simulator to provide the irulation by four small fans installed in the irulation hamber. Transition of a vortex from a one-ell to two-ell struture ourred throughout the whole onvergene layer in their simulator. Haan et al. (8) developed a large laboratory simulator with guide-vanes at the top to make the translation of the tornadoes reproduible. Most reently, Tari, P.H. et al. () quantified both the mean and turbulent flow fields for a range of swirl ratios spanning from F to F sale by using the Partile Image Veloimetry (PIV) method. However, in view of the limitation of observation methods and the extremely ompliated flow fields near the ground, it is diffiult to make detailed three-dimensional measurements in the boundary layer whih has been universally believed to be the most important region in the tornado-like vorties. Reently, with the advanement in the omputer tehnology, many numerial studies have been onduted. Nolan and Farrell (999) explored the dynamis of axisymmetri tornado-like vorties. The internal swirl ratio as well as the vortex Reynolds number was defined. D.C
Shanghai, China; September -6, Lewellen et al. () explored some tornado strutures being expeted to our in nature. They defined the loal orner swirl ratio and proposed the existene of a ritial swirl ratio, at whih the largest swirl veloity ours very lose to the ground. Ishihara et al. () used LES turbulent model to simulate the flow fields of two types of tornado-like vorties and validated the model by omparing with laboratory simulators. The formation of one-ell and two-ell type vorties were investigated by examining axisymetri time averaged Navier-Stokes equations. However, among all the simulators the definition for the swirl ratio is not uniform and varies from one to another. In this study, a numerial model representing the Ward-type tornado simulator is built and four typial types of tornado vorties are examined. The details of this model and the ase settings are introdued in setion inluding its dimension, grid distribution and boundary onditions. In setion 3 the three-dimensional flow fields as well as the fore balane is provided. Setion 4 proposes a universal definition of swirl ratio to unify the researhes. NUMERICAL MODEL. Governing equations The governing equations employed in LES model are obtained by filtering the time-dependent Navier-Stokes equations as follows: u i () x i ui P ij ( ui ) ( uiuj) () t xj x j x j xi xj where u i and ~ p are filtered mean veloity and filtered pressure respetively. ρ is density, τ ij is subgrid- sale stress and is modeled as follows: ij t S ~ u u i j ij kk ij, S ij (3) 3 x j x i where μ t is subgrid-sale turbulent visosity, and S ~ ij is the rate-of-strain tensor for the resolved sale. Smagorinsky-Lilly model is used for the subgrid-sale turbulent visosity, ~ ~ ~ t L s S /3 Ls Sij Sij, Ls min, C s V (4) where L s is the mixing length for subgrid-sales, κ is the von Karman onstant,.4, C s is Smagorinsky onstant, δ is the distane to the losest wall and V is the volume of a omputational ell. In this study, C s is determined as.3 based on Oka and Ishihara (9). When a wall-adjaent ell is in the laminar sublayer, the wall shear stress is obtained from the laminar stress-strain relationship as follows: u u y u (5) If the mesh annot resolve the laminar sublayer, it is assumed that the entroid of the walladjaent ells fall within the logarithmi region of the boundary layer, and the law-of-the-wall is employed:
Shanghai, China; September -6, u uy ln E (6) u k where u is the filtered veloity tangential to wall, u τ is the frition veloity and the onstant E is 9.793.. Configurations and solution sheme In this study, a Ward-type simulator (Ward et al. 97) is hosen and numerially simulated. The onfigurations of the numerial model are shown in Figure (a). Two signifiant geometry parameters are the height of the inlet layer, h, and the radius of the updraft hole, r o, whih are mm and 5mm respetively. The veloity profiles at the inlet are speified as below: n z Urs U z (7) V U tan( ) rs rs where, U rs and V rs are radial veloity and the tangential veloity at r=r s, n equals to 7, the referene veloity U and the referene height z are set to.4m/s and.m respetively through mathing the veloity profile in the previous study by Ishihara et al. (), and θ is the degree of the inflow angle. Considering the axisymmetry of tornado-like vortex, an axisymmetri topology method is adopted, see Figure (b). With an intent to investigate the turbulent features quantitatively in the viinity of the enter and the region near the ground, very fine mesh is onsidered in the onvergene region. The total mesh number is about 7.8 6. Table summarizes the parameters for the mesh and the system of the PC luster used in this study. Z X Y a b Figure. Geometry(a) and mesh(b) of the model.
Shanghai, China; September -6, Table. Parameters for the mesh and the system of PC luster Mesh size in the radial diretion.~5.mm Mesh size in the vertial diretion.~5.mm Mesh number 784 CPU Athlon 64 Proessor3,.GHz Number of nodes 8 CPU time for the ase of S=.44 5h Finite volume method is used for the present simulations. SIMPLE (semi-impliit pressure linked equations) algorithm is employed for solving the disritized equations (Ferziger and Peri, ). The pressure at inlet of the onvergene region is set to zero, and veloity at the outlet (,, W ) is given as (,, 9.55m/s) to generate upward flow in the tornado. Three veloity omponents and the pressure are set as zero for the initial onditions..3 Swirl ratio definitions and Case settings In both the laboratory and the numerial simulators, a orrelation has been found between the vortex struture and the swirl ratio. Various definitions for the swirl ratio have been proposed in the previous studies. The swirl ratio has historially been defined as the ratio of angular momentum to radial momentum in the vortex, and expressed in the following forms: r tan Sout, Sout and Sout (8) Qh Qa a where, Γ is the irulation at the outer edge of the onvergene region, Γ =πr s hv, and a is the aspet ratio, a=h/r. In ase the irulation is imposed by using guide vanes instead of rotating sreen, the ratio of the irulation rate to the volume flow rate an easily be replaed by tanθ, where θ is the angle of the guide vanes. In ISU tornado simulator by F. L. Haan Jr.(8) the swirl ratio is speifially modified as: r r( rv ) rv S (9) Qh Q Q in whih, the irulation, Γ, is estimated using the multipliation of the maximum tangential veloity, V, and r, Γ =πr hv, r is radius of maximum tangential veloity, V, in the quasiylindrial region. D.C. Lewellen() proposed a loal orner flow swirl ratio, S. The speifi form of the loal orner flow swirl ratio is: * * r S (-a) in whih, r * is the harateristi length sale, alulated as r * * /V, π * is the irulation per unit height in the outer region expressed as π * =πv r, is the total depleted irulation flux flowing through the orner flow region, expressed as: r W( r, z) d ( r, z) rdr (-b) where, Γ d is the depleted angular momentum and defined as Γ d = Γ -Vr, r is the radius safely outside of the upper-ore region, z is the height just above the orner flow.
Shanghai, China; September -6, Table. Case settings and aompany tornado vortex parameters. Case θ ( o ) S out S S Re Q(m 3 /s) V (m/s) r (m) Case 46.8.4..7.6 5.3.7.4 Case 58..6.6.59.6 5.3 9.84.4 Case3 64.9.8..36.6 5.3 9..35 Case4 69.4.3.93.6 5.3 9.6.47 Case5 76..5.34 4.6.6 5.3.99.54 Case6 79.4.69 5.39.6 5.3.35.73 Case7 8..7.6 6.74.6 5.3 4.6.84 Case8 83.5 3.3.58 7.96.6 5.3 5.98.5 Case9 84.4 3.8.44 8.89.6 5.3 8.6. Previous 6..65.8 3.5.6 5.3 8.33.3 Notes: Case, Case, Case4 and Case9 are four typial types of tornado onfiguration and are hosen for detailed flow field analysis. Previous ase is the simulation arried out by Ishihara (). In this study, the swirl ratio is inreased through inreasing the inflow angle. Nine ases are alulated systematially. The ase settings as well as tornado vortex parameters for eah ase are illustrated in Table, in whih Re is the Reynolds number defined as Re=W D/υ, D=r. It an be found the swirl ratio S out tends to have a larger value ompared with S. On the other hand, the loal orner swirl ratio S shows the maximum value for the same orner flow pattern. 3 FLOW FIELDS CHARACTERISTICS It is neessary to make the air flow visible and evaluate the vortex in a qualitative manner. For this purpose virtual water vapor is seleted as the visualizing substane injeted from the bottom of the model. The partiles are not released until the flow fields are in the quasi-steady stage to eliminate the effet of the transit field solution. As the value of the swirl ratio is inreased, the vortex goes through various stages, as depited in Figure. When S=., we find the entral ore to have a smooth, laminar appearane. The ore extends upward from the surfae to the high elevation spreading radially slightly with height, shown in Figure (a). For S=.6, a vortex breakdown ours where the flow transitions from a tight, laminar vortex to a broader, turbulent state, see Figure (b). At S=.3, the radius of the vortex ore inreases and the altitude of the breakdown dereases, as shown in Figure (). The vortex breakdown is just above the boundary layer. A still further inrease in swirl ratio to S=.44 results in the breakdown being fored further toward the surfae layer, see Figure (d). The ore of the vortex expands substantially, leaving a relatively alm inner subore. Conurrent with the expansion of the ore the inner downflow penetrates to the lower surfae, and in this partiular snapshot, a family of several seondary vorties rotating about the main vortex is evident. 3. Mean flow fields Quantitative analysis an be ahieved by examining the distributions of the mean veloity omponents. In the following disussion, the maximum tangential veloity in the ylostrophi balane region, V, will be used to normalize the flow fields. The radial distane is normalized by the ore radius of the tornado vortex in ylostrophi balane region, r.
Shanghai, China; September -6, Figure. Flow visualization by injeting water vapor from the ground for four typial types of tornado-like vorties, (a) weak vortex, S=., (b) vortex breakdown, S=.6, () vortex touh-down, S=.3, (d) multi-vortex, S=.44..5 z=.ro z=.4ro z=.7ro z=.ro.5 z=.ro z=.4ro z=.7ro z=.ro V/V V/V.5.5 3 4 3 4 a b.5 z=.ro z=.4ro z=.7ro z=.ro.5 z=.ro z=.4ro z=.7ro z=.ro V/V V/V.5.5 3 4 3 4 d Figure 3. Radial profiles of the normalized tangential veloity for four typial types of tornado-like vorties, (a) weak vortex, S=., (b) vortex breakdown, S=.6, () vortex touh-down, S=.3, (d) multi-vortex, S=.44.
Shanghai, China; September -6, The radial profiles of the mean tangential veloity, V, versus nondimensional radial distane are shown in Figure 3(a). For the very low swirl ratio ase, S=., the mean tangential veloity field is apparently one dimensional exept the layer very near the ground. The ore radius, R, defined based on the loation of the maximum tangential veloity at eah elevation, is almost onsistent. For S=.6, the swirl overshoot appears at the surfae layer with maximum tangential veloity being.6 times of V, and the ore radius inreases from.5r to.r with height forming a funnel shape. Further inreasing the swirl ratio to the stages of touh-down and multi-vortex, the ratio of maximum swirl veloity, V max, to V is nearly a onstant, varying in between.3 to.5, and the radial loation of V max hanges very slightly, holding about.5r. 3. Fore balanes analysis The ontributions from eah term in Navier-Stokes equation an be alulated by fore balanes analysis. Ishihara, T. () investigated the fore balanes of two typial stages by using the time-averaged axisymmetri Navier-Stokes equations. However, a systemati ross omparison for the fore balanes in various types of vorties is limited and deserved to be studied. The time-averaged radial Navier-Stokes equation an be expressed as: U U V P u uw v u U W Du () r z r r r z r r The left hand side onsists of the radial advetion term, A ru, the vertial advetion term, A zu, as well as the entrifugal fore term, C r. The right hand side of the equation is the radial pressure 5 -Aru -Azu Cr Pr Tu 5 -Aru -Azu Cr Pr Tu -5-5 -.5.5 -.5.5 a b 5 -Aru -Azu Cr Pr Tu 5 -Aru -Azu Cr Pr Tu -5-5 -.5.5 -.5.5 d Figure 4. Radial fore for four typial types of tornado-like vorties, (a) weak vortex, S=., (b) vortex breakdown, S=.6, () vortex touh-down, S=.3, (d) multi-vortex, S=.44.
Shanghai, China; September -6, gradient term, P r, turbulent fore term, T u, and the diffusion term, D u. The diffusion term, D u, in the equation is small enough to be ignored ompared with the other terms. u,v,w are root mean squares of the radial, tangential and vertial veloities. Due to the slight hange for the height of maximum tangential veloity, the terms in the radial momentum equations are omputed at z=.r o as a funtion of for all the four stages, as shown in Figure 4. Examination of Figure 4(a) reveals that turbulene plays little role for weak vortex stage in the radial momentum balane. The entrifugal term and pressure gradient term are the signifiant portion of the total balane. Inreasing the swirl to S=.6, the flow evolves from laminar vortex to a turbulent state, followed by a signifiant hange of the radial balane, as shown in Figure 4(b). The priniple balane is in between the entrifugal term, pressure gradient term, turbulent term and vertial advetive term. The emergene of the turbulent term is the manifestation of the unsteadiness of the flow fields. Figure 4() displays the terms in the radial balane for the ase of vortex touh-down. The entrifugal term is mainly balaned by the pressure gradient term as well as the vertial advetive term. However, different with the state of vortex breakdown, in some region the vertial advetion term beomes more important than the pressure gradient term. Radial fore balane for multi-vortex is presented in Figure 4(d), whih is almost oinident with that for vortex touh-down. 4 PERFORMANCE OF LOCAL CORNER SWIRL RATIO A pitorial approah is adopted to show how the surfae intensifiation and the shape of the vorties hange with the swirl ratio S, as illustrated in Figure 5, where U min is the minimum averaged radial veloity, V max is the maximum averaged tangential veloity, W max is the maximum vertial veloity, r vmax and h vmax are the radius and height of the loation of the maximum tangential veloity respetively. The parameters V max /V, -U min /V max, W max /V max and the ratio of r vmax to h vmax are examined. The results of this study will be ompared with the previous laboratory-sale numerial study by Ishihara et al. () and the full sale numerial study by Lewellen et al. (). It is worth to be mentioned that the methods to obtain the angular momentum are different from one to another. In the previous study by Ishihara et al. (), the azimuthal momentum of the inflow is imposed by guide vanes, while in the present 8 ases the irulation is obtained diretly from the veloity profile at the inlet boundary. For the full sale numerial model by Lewellen et al. (), the boundary ondition is obtained from an inner nest of a thunderstorm simulation. The ratio of the maximum averaged swirl veloity, V max, to the maximum averaged swirl veloity in the upper ylostrophi region, V, as a funtion of the loal orner swirl ratio is demonstrated in Figure 5(a). Examining the set of present ases, the ratio inreases sharply from the very low swirl ratio until S equals to around.6 where the pattern of vortex breakdown ours and the distint peak ratio reahes to about.7. For inreasing the swirl ratio, the ratio of V max to V dereases moderately and at last beomes almost a onstant varying the values in between.3 and.5. Figure 5(b) shows the ratio -U min /V max as a funtion of the loal orner swirl ratio. This ratio inreases suddenly from the state of weak vortex, however, it is obvious that, exept the very low swirl ases, -U min /V max is insensitive to the swirl ratio and all the data are sattered about a entral value.65. This near onsisteny is the indiation of the dependeny between the lowlevel radial overshoot and the swirl overshoot.
Shanghai, China; September -6, 4 /hvma3 Present Previous LEWELLEN.8 Present Previous LEWELLEN V max / V -U max / V max.6.4. 4 6 8 4 6 8 S S 3 a 8 b W max / V max Present Pervious LEWELLEN vmaxpreviousr6 4 Present xs 4 6 8 S 4 6 8 d Figure 5. Summery surfae intensifiation and the geometry of tornado vorties as a funtion of loal orner swirl ratio, (a) the ratio of V max /V, (b) the ratio of -U min /V max, () the ratio of W max /V max and (d) the aspet ratio of r vmax /h vmax. The ratio W max /V max shows the maximum value, about.9, at the stages with very low swirl ratios, and dereases with inreasing the swirl ratio, as demonstrated in Figure 5(). The maximum vertial veloity is larger than the maximum tangential veloity until the vortex reahes to the touh-down state, after whih the near onsisteny for the ratio W max /V max is observed with a entral value of about.4. A vortex aspet ratio defined as the ratio of r vmax to h max is applied to evaluate the struture of the flow in the vortex orner. As observed in Figure 5(d), the vortex aspet ratio inreases linearly with S having a slope of about.7. The values of V max /V, -U min /V max and W max /V max as a funtion of loal orner swirl ratio obtained in Lewellen et al. () and Ishihara et al. () are also shown for omparison. Even though the present, previous and Lewellen s ases use different numerial models to generate the vorties, it is obvious that the results from different models exhibit the universally same tendenies. Therefore it an be argued that it is reasonable to universalize the researhes by using the loal orner swirl ratio.
Shanghai, China; September -6, 5 CONCLUSIONS The flow fields as well as the fore balanes of tornado-like vorties have been investigated by using the LES turbulent model in this study. Following summarizes the onlusions.. The visualized flow fields by the injeted water fog from the ground suessfully show the evolution from a single-elled vortex into a two-ell vortex onfiguration ontaining multiple subsidiary vorties.. The maximum normalized tangential veloity ours at the stage of vortex breakdown, after whih the maximum tangential veloity almost holds as a onstant and the normalized radial loation of V max hanges very slightly. Turbulene plays little role for weak vortex stage in the radial momentum balane. From the stage of vortex breakdown the fore balane hanges dramatially and the effets of turbulene emerge. 3. The values of V max /V, -U min /V max and W max /V max obtained from different simulators are plotted as a funtion of loal orner swirl ratio and show good omparison. The vortex aspet ratio r vmax /h rvmax shows linear relationship with the loal orner swirl ratio having a slope of about.7. Based on the onsisteny provided by the loal orner swirl ratio, it is proposed to universalize the researhes. REFERENCES Ferziger, J. and Peri, M., Computational method for fluid dynamis, 3rd Edition,,, Springer. Haan, F.L., Sarkar, P.P., Gallus, W.A., Design, onstrution and performane of a large tornado simulator for wind engineering appliations, Engineering Strutures, 8, Vol.3, pp.46-59. 3 Ishihara, T., Oh, S., and Tokuyama, Y., Numerial study on flow fields of tornado-like vorties using the LES turbulene model, Journal of Wind Engineering and Industrial Aerodynamis,, Vol.99, pp.39-48. 4 Lewellen, D.C., Lewellen, W.S., and Sykes, R.I., Large-eddy simulation of a tornado s interation with the surfae, Journal of the Atmospheri Sienes, 997, Vol.54, pp.58-65. 5 Lewellen, D.C., Lewellen, W.S., and Xia. J., The influene of a loal swirl ratio on tornado intensifiation near the surfae, Journal of the Atmospheri Sienes,, Vol.57, pp.57-544. 6 Mitsuta,Y. and Monji, N., Development of a laboratory simulator for small sale atmospheri vorties, Natural Disaster Siene, 984, Vol.6, pp.43-54. 7 Nolan, D.S. and Farrell, B.F., The struture and dynamis of tornado-like vorties, Journal of the Atmospheri Sienes, 999, Vol.56, pp.98-936. 8 Oka,S. and Ishihara,T., Numerial study of aerodynami harateristis of a square prism in a uniform flow. Journal of Wind Engineering and Industrial Aerodynamis, 9, Vol.97, pp.548 559. 9 Tari, P.H., Gurka, R., Hangan, H.. Experimental investigation of tornado-like vortex dynamis with swirl ratio: The mean and turbulent flow fields, Journal of Wind Engineering and Industrial Aerodynamis,, Vol.98, pp.936-944. Ward, N.B., The Exploration of Certain Features of Tornado Dynamis Using a Laboratory Model, Journal of the Atmospheri Sienes, 97, Vol.9, pp.94-4.