Numerical study on dynamics of a tornado-like vortex with touching down by using the LES turbulence model

Transcription

1 Wind and Strutures, Vol. 9, No. (4) - DOI: Numerial study on dynamis of a tornado-like vortex with touhing down by using the LES turbulene model Takeshi Ishihara and Zhenqing Liu a Department of Civil Engineering, Shool of Engineering, The University of Tokyo, 7-3- Hongou, Bunkyo-ku, Tokyo , Japan (Reeived September,, Revised Marh, 4, Aepted May 8, 4) Abstrat. The dynamis of a tornado-like vortex with touhing down is investigated by using the LES turbulene model. The detailed information of the turbulent flow fields is provided and the fore balanes in radial and vertial diretions are evaluated by using the time-averaged axisymmetri Navier-Stokes equations. The turbulene has slightly influene on the mean flow fields in the radial diretion whereas it shows strong impats in the vertial diretion. In addition, the instantaneous flow fields are investigated to larify and understand the dynamis of the vortex. An organized swirl motion is observed, whih is the main soure of the turbulene for the radial and tangential omponents, but not for the vertial omponent. Power spetrum analysis is onduted to quantify the organized swirl motion of the tornado-like vortex. The gust speeds are also examined and it is found to be very large near the enter of vortex. Keywords: dynamis of tornado-like vortex with touhing down; LES; turbulent flow fields; fore balanes; organized swirl motion; power spetrum; gust speed. Introdution Tornadoes are vorties with strong three-dimensional flow fields and ause severe damages. Wind resistant design of strutures requires proper onsideration of tornado-indued wind loads and tornado-borne missiles, whih requires detailed information of the three-dimensional flow fields of tornadoes. In laboratory simulations, Ward (97) first developed a tornado simulator with a fan at the top to generate updraft flow and guide-vanes near the floor to provide angular momentum, and sueeded in generation of many types of tornado-like vorties observed in nature. Wan and Chang (97) measured the radial, tangential and vertial veloities with a three-dimensional veloity probe and provided a feasible supplement to field measurements of the natural tornado. Baker (98) made the veloity measurements in a laminar vortex boundary layer by employing hot-film anemometry. Mitsuta and Monji (984) modified the simulator so that the rotation is given at the top of the simulator, and showed the ourrene of the maximum tangential veloity near the ground in the two-ell type vortex. Haan et al. (8) developed a large laboratory simulator with Corresponding author, Professor, ishihara@bridge.t.u-tokyo.a.jp a Ph.D. Student, liu@bridge.t.u-tokyo.a.jp Copyright 4 Tehno-Press, Ltd. ISSN: 6-66 (Print), (Online)

2 Takeshi Ishihara and Zhenqing Liu guide-vanes at the top to generate vortex. In their study, the measurements of the flow struture in the vortex were validated by omparing with mobile Doppler radar observations of two major tornadoes. Matsui and Tamura (9) onduted the veloity measurements for tornado-like vortex generated by a Ward-type simulator with Laser Doppler Veloimeter (LDV). Reently, Tari et al. () quantified both the mean and turbulent flow fields for a range of swirl ratios by using the Partile Image Veloimetry (PIV) method. They showed a ritial swirl ratio, where the turbulent vortex touhes down and the turbulent prodution approahes maximum values. In addition, they argued that lose to the ground, the turbulene is responsible for the damage assoiated to tornado events. However, in laboratory simulations, it is diffiult to make detailed three-dimensional measurements due to the strong turbulene near the surfae and the orner of the vortex. Numerial modeling paves the way for better understanding of tornadoes. Wilson and Rotunno (986), Howells et al. (988) and Nolan and Farrell (999) used axisymmetri Navier-Stokes equation in ylindrial oordinates to investigate the flow fields of the tornado-like vortex as well as the dynamis in the boundary layer and orner regions. Kuai et al. (8) used the k-ε model to study parameter sensitivity for the flow fields of a laboratory-simulated tornado. They proposed that the numerial approah an be used to simulate the surfae winds of a tornado and to ontrol ertain parameters of the laboratory simulator. However, when the vortex experienes a breakdown, the flow will be no longer axisymmetri, whih means that dynamis of the vertex annot be simulated with a two-dimensional axisymmetri model. Lewellen et al. (997,, 7) used three-dimensional LES turbulene model to examine several types of tornado-like vortex, the influene of the translation speed as well as the interation with the surfae roughness. They proposed to use the loal swirl ratio to identify the tornado strutures, and it is found when S., a sharp vortex breakdown aps a strong entral jet forming a touhing-down struture. The presene of vortex touhing-down results in the largest swirl veloity ourring near the ground. Ishihara et al. () investigated the detailed orner flow patterns of tornado-like vorties by using the LES turbulene model for two typial swirl ratios aording to weak vortex and vortex touhdown, and the mehanism of the flow field formation was revealed by evaluating the axisymmetri time averaged Navier-Stokes equations. Most reently a large eddy simulation about the roughness effets on tornado-like vorties was arried out by Natarajan and Hangan (). The effets of the swirl ratio on the turbulent flow fields of tornado-like vorties were investigated by Liu and Ishihara (), in whih, similar as the study by Lewellen et al. (), the vortex touhing down status also ours at S.. The studies by Lewellen et al. (, 7) give similar overshooting profiles near the ground as for tangential and radial veloities as well as the large wind flutuations near the enter at the stage of vortex touhing down, the mehanism of the large flutuations, the spetra of flutuating veloities and gust speeds in tornado-like vorties with touhing-down have not been made lear. In this study, a omprehensive study of the tornado-like vortex with touhing down is performed by using LES turbulene model to shed light on the dynamis of a tornado-like vortex in the orner region where is of interest for engineering appliations. Numerial model and onfigurations of a numerial Ward-type tornado simulator are desribed in setion, inluding its dimension, grid distribution and boundary onditions. Setion 3 aims to larify the mean and turbulent harateristis of the vortex and to examine the fore balanes and the turbulent ontribution to the mean flow by the axisymmetri time-averaged Navier-Stokes equations. Finally, the instantaneous flow field and the spetrum analysis are employed to examine the organized

3 Numerial study on dynamis of a tornado-like vortex with touhing down motion in the tornado-like vortex with touhing-down and the gust speeds are investigated to provide valuable information for the design purposes in setion 4.. Numerial model In this study, large eddy simulation (LES) is adopted to simulate the tornado-like vortex, in whih large eddies are omputed diretly, while the influene of eddies smaller than grid spaing are modeled. Boussinesq hypothesis is employed and standard Smagorinsky-Lilly model is used to alulate the subgrid-sale (SGS) stresses.. Governing equations and boundary onditions The governing equations are obtained by filtering the time-dependent Navier-Stokes equations in Cartesian oordinates (x, y, z) and expressed in the form of tensor as follows u~ i () x u~ u ~ i iu ~ j u ~ i ~ p t x j x j x j xi j ij x where, u ~ i and ~ p are filtered veloities and pressure respetively, is the visosity, is air density, is SGS stress and is modeled as follows where ij u ~ ~ ~ u ~ i j ij tsij kkij, Sij (3) 3 x j xi t denotes SGS turbulent visosity, and S ~ ij j () is the rate-of-strain tensor for the resolved sale, ij is the Kroneker delta. Smagorinsky-Lilly model is used for the SGS turbulent visosity ~ ~ ~ ; min, 3 t LS S LS SijSij LS d CSV (4) in whih, L S denotes the mixing length for subgrid-sales, is the von Kármán onstant,.4, d is the distane to the losest wall and V is the volume of a omputational ell. In this study, Smagorinsky onstant, C S, is determined as.3 based on Oka and Ishihara (9) For the wall-adjaent ells, the wall shear stresses are obtained from the laminar stress-strain relationship in the laminar sublayer u ~ u y (5) u

4 Takeshi Ishihara and Zhenqing Liu Fig. The geometry of the model When the entroid of the wall-adjaent ells falls within the logarithmi region of the boundary layer, the law-of-the-wall is employed as follows u ~ u ln E k u y where u ~ is the filtered veloity tangential to wall, y is the distane between the enter of the ell and the wall, u is the frition veloity, and the onstant E is (6). Configurations of the numerial tornado simulator In this study, a Ward-type simulator is hosen and the onfigurations of the model are shown in Fig.. The angular momentum of the flow entering into the onvergene region is obtained by 4 guild vanes mounted on the ground at radius of 75 mm. The height of the guide vanes is mm and the width is 3 mm. The angle of the guild vanes,, is 6 o. The height of the inlet layer, h, and the radii of the updraft hole, r, are mm and 5 mm respetively. The total outflow rate Q r t W is held as a onstant.3 m 3 /s, where rt is the radius of the exhaust outlet and W is the veloity, 9.55 m/s, at the outlet. The dominant parameter determining the struture of a tornado-like vortex is identified as the swirl ratio and expressed as S tan a, where a denotes the internal aspet ratio and equals to h r. The loal swirl ratio, S, defined by Lewellen et al. (), is alulated as.. Another dimensionless parameter is the Reynolds number, W D, where D is the diameter of the updraft hole. The maximum mean R e tangential veloity, V, in the ylostrophi balane region is 8.33 m/s, and the radius, r, where the maximum mean tangential veloity ours, is 3.6 mm.

5 Numerial study on dynamis of a tornado-like vortex with touhing down.3 Numerial sheme Finite volume method is used for the present simulations. The seond order entral differene sheme is used for the onvetive and visosity term, and the seond order impliit sheme for the unsteady term. SIMPLE (semi-impliit pressure linked equations) algorithm is employed for solving the disretized equations (Ferziger and Peri ). Table Parameters used in this study Angle of the guild vanes: 6o Height of the inlet layer: h mm Radius of the updraft hole: r Reynolds number: Re rw Non-dimensional time step :.63 5 t W r.3 5 mm Mesh size in the radial diretion. 6. mm Radius of the exhaust outlet: rt mm Mesh size in the vertial diretion. 5. mm Veloity at the outlet: W 9.55 m/s Mesh number 6497 Total outflow rate: Q rt W.3 m /s Maximum mean tangential veloity: V 8.33 m/s Internal aspet ratio: a h r.33 Radius of the ore: r 3.6 mm Swirl ratio: S tan a.65 3 Z X Y (a) Bird s view Y Z Z Y X (b) Side view () Top view Fig. Mesh of the numerial model X

6 Takeshi Ishihara and Zhenqing Liu At the inlet of the onvergene region the pressure is set as zero. Uniform veloity boundary ondition representative of the exhausting fan is speified at the outlet surfae, where the radial, tangential and vertial veloities are, and 9.55 m/s respetively. Non-slip boundary ondition is applied at the bottom and the wall of the simulator. In most of the region, the wall-adjaent ells are in the laminar sublayer. The maximum of y+ is 6 in the ore region of the vortex. There are 7 grid points below the peak of mean radial veloity and 5 grid points below the peak of mean tangential veloity to make sure the boundary layer ould develop properly. A non-dimensional time step of.3 is adopted, alulated as tw D, where t is the time step. The numerial solution was arried out to 3s and the first s data were removed to eliminate the transit result. The data from s to 3s were applied to obtain the statistial information of the flow fields in the tornado-like vortex. The relative errors derease with inreasing in alulating time. For s, the relative error of the maximum mean tangential veloity in the ylostrophi balane region beomes less than %. Considering the axisymmetry of tornado-like vortex, a novel axisymmetri topology method is adopted, as shown in Fig.. With an intent to investigate the turbulent features in the viinity of the enter and the region near the ground, a fine mesh is onsidered in the domain of the onvergene region, where 68 nodes in the radial diretion and 5 nodes in the vertial diretion are used, and the minimum size of the mesh are about mm in the radial diretion and mm in the vertial diretion. The spaing ratios in the two diretions are less than. in order to avoid a sudden hange of the grid size. The total mesh number is about 6 5. Table summarizes the parameters used in this study. 3. Turbulent harateristis The turbulent harateristis of the tornado-like vortex with touhing-down are summarized in this setion. First, the profiles of the mean and the flutuating veloities and pressure, as well as the turbulent kineti energy (TKE) and the Reynolds shear stress uw are disussed. The alulation of the statistial values at eah position is performed by averaging the values at the same radius and height over twelve azimuthal angles. Then, the fore balanes in radial and vertial diretion are evaluated. 3. Turbulent flow fields Fig. 3 shows the omparison of the flow fields in the laboratory tests by Matsui and Tamura (9) and that in the numerial tornado simulator. Small partiles are injeted from the bottom in the numerial model with the objetive of simulating the smoke used in the experiments. The flow fields are very turbulent and the ore radii are almost same with eah other. The averaged flow fields on the horizontal ross-setion of z. r and vertial ross-setion of y r are shown in Figs. 4(a) and 4(b) respetively. The axisymmetri flow pattern an be observed learly in the horizontal ross-setion with the enter of the flow fields oiniding with that of the simulator. In the outside region, the signifiant radial omponent in addition to the tangential omponent indiates the feature of spiral motion whih an be found from the streamlines superimposed on Fig. 4(a). On the vertial ross-setion, at the enter the flow moves downward, and near the ground a radial jet penetrates to the axis then turns upward. The radial jet

7 Y/r Z/r Numerial study on dynamis of a tornado-like vortex with touhing down and the downward jet break away at the stagnation point, r and z. 3r, and a irulation zone is generated at the interfae between the near-surfae onial vortex and the aloft ylindrial vortex. r r (a) Experiment (b) Simulation Fig. 3 (a) Flow fields visualized by the smoke injeted from bottom of the laboratory simulator and (b) flow fields visualized by small partiles injeted from the bottom of the numerial simulator (a) X/r X/r. (b) y r z r Fig. 4 Vetors of mean veloities on ross-setions of (a) z r. and (b) y r

8 Takeshi Ishihara and Zhenqing Liu The mean and flutuating veloity profiles normalized by V at three elevations z. r, z. 5r and z. r are depited in Fig. 5. Fig. 5(a) presents the mean radial veloity profiles. Major portion of the radial inflow is onentrated in a thin layer next to the ground whih an be found in the profile at the elevation z. r. The onentration of the radial inflow will be explained by the fore balane analysis in setion 3.. At the r r, the maximum radial inward veloity reahes to about.7v. The radial veloity dereases with the dereasing in radial distane and hanges its sign near the enter. At the height of z. 5r, the radial veloity shows positive sign in the ore region, r r, whih an be explained like that the entral downward flow meets upward flow and pushes the radial flow outward. Moving to high elevation, z. r, the flow reahes to the ylostrophi balane, whih means the entrifugal fore due to the motion of the flow balanes with the pressure gradient, and the diretion of the motion of the flow should perpendiular to the diretion of the pressure gradient, therefore the flow an only moves tangentially or vertially. This an be learly found in the profile of the radial veloity at z. r where the radial veloity is around. Fig. 5(b) shows the root mean square (r.m.s) of the flutuating radial veloity, whih inreases with derease in the radial distane at most elevations and shows peak in the enter. It is noteworthy that, at z. 5r, u is almost a onstant in the ore region, whih an be attributed to the mixing of the flow. At the enter the r.m.s radial veloity shows large value while the mean radial veloity is zero. This will be explained in the following disussion about the dynamis of the tornado. Normalized tangential veloities in respet to non-dimensional radial loation are presented in Fig. 5(). It is obvious that overall the maximum tangential veloity happens at r r, exept the loations lose to the ground, where the tangential veloity inreases to about.4 V at r. 5r. This inrease in the tangential veloity is important for wind resistant design, sine most of the engineering strutures exist in the surfae layer. Comparing the profiles of tangential veloity at high elevations, a shape similarity is revealed. The tangential veloity is initially zero at the enter then inreases to the maximum, for further inreasing in radial distane it dereases. The profile of tangential veloity at z. 67r is also plotted with the onsideration of omparing the results by Matsui and Tamura (9) and it shows good agreement with the experiment. The profiles of the r.m.s tangential veloity, v, is illustrated in Fig.5(d). Similar with the radial flutuations, at most elevations the v shows the largest value at the enter, while, at z. 5r, v is almost a onstant along the radius from r to r. 4r. Same with the radial omponent, the generation of turbulene for the tangential omponents is ompletely different from that of the general turbulene whose soure is mainly the veloity gradients as the fully developed boundary layer winds. Fig. 5(e) shows the mean vertial veloities as a funtion of radial distane. The maximum axial veloity,.6v, ours at the enter with a height of. r. This large axial veloity is attributed to the radial jet hanging its diretion to upward here. In the upper region, z. 5r, the vertial veloity first inreases then dereases with dereasing in radial distane and the downward vertial veloity is observed at the enter of the vortex. The normalized vertial veloities from the laboratory by Mitsuta and Monji (984) are also plotted and the predited veloities show satisfatory agreement with the measured ones at z. 3r. The profiles of w, resembles those of

9 Numerial study on dynamis of a tornado-like vortex with touhing down the radial and tangential stresses in spite of the relatively smaller values, as shown in Fig. 5(f). Near the ground, z. r, w inreases as derease in the radial distane. The highest value of the r.m.s vertial veloity,.5v, ours at the enter and another relatively large r.m.s vertial veloity ours at z. 5r, r. 4r, where the strong mixing effets an be observed. Z=.r Z=.r.5 Z=.5r.5 Z=.5r Z=.r Z=.r U/V u/v r/r (a) Mean radial veloity.5.5 r/r (b) r.m.s of radial veloity Z=.r Z=.67r.5 Z=.5r Z=.r Matsui z=.67r.5 Z=.r Z=.5r Z=.r V/V v/v r/r r/r () Mean tangential veloity.5 (d) r.m.s of tangential veloity Z=.r Z=.3r Z=.r Z=.5r Monji & Mitsuta Z=.3r.5 Z=.5r Z=.r Z=.r W/V.5 w/v r/r (e) Mean vertial veloity (f) r.m.s of vertial veloity Fig. 5 Normalized mean veloity and r.m.s of the flutuating veloities at three vertial positions. (a) mean radial veloity, (b) r.m.s of radial veloity, () mean tangential veloity, (d) r.m.s of tangential veloity, (e) mean vertial veloity, and (f) r.m.s of vertial veloity r/r

10 P/V p/v k/v uw/v Takeshi Ishihara and Zhenqing Liu.5.5 Z=.r Z=.5r. Z=.r Z=.5r Z=.r Z=.r r/r r/r (a) Turbulent kineti energy k (b) Reynolds stress uw Fig. 6 Horizontal distribution of (a) turbulent kineti energy k and (b) Reynolds stress uw Z=.r Z=.r Z=.5r Z=.5r - Z=.r.5 Z=.r r/r r/r (a) Mean pressure (b) r.m.s of the pressure flutuation Fig. 7 Horizontal distribution of (a) mean pressure and (b) r.m.s of the pressure flutuation The horizontal distributions of the normalized turbulent kineti energy, k, are shown in Fig. 6(a). The maximum turbulent kineti energy is about.8 ourring at the axis with a height of. r, and the generation of the turbulent kineti energy omes mainly from the radial and tangential flutuating veloities. The zero values for the mean radial and tangential veloities along the entral line lead to the onlusion the turbulene in the enter plays the dominant role for the damages assoiated to the tornado events. Another important variable used to analyze turbulene is the Reynolds shear stress, thus the profiles of the normalized Reynolds shear stress, uw are also presented in Fig. 6(b). Compared with the normal stresses, the shear stress is less than the normal stresses with one order of magnitude. In addition, at the entral line the shear stress is zero, while the normal stresses show maximum values. Fig. 7(a) illustrates the variations of the normalized mean pressure. Different with the veloity omponents, the profiles of the pressure show almost same shapes at various elevations. As losing

11 Numerial study on dynamis of a tornado-like vortex with touhing down to the entral axis, signifiant pressure drop is observed. The horizontal distributions of the r.m.s for the pressure flutuation, p, are illustrated in Fig. 7(b), whih shows the maximum value of rms.45 at the height of. r, and the magnitude is muh smaller than that of the mean pressure at all loations. 3. Fore balanes in radial and vertial diretion The radial and vertial fore balanes in tornado-like vortex are investigated by using the time-averaged axisymmetri Navier-Stokes equations as shown by Ishihara et al. (). The sub-terms of the turbulent are also analyzed in detail to examine how the turbulene influenes the mean flow fields. The time-averaged Navier-Stokes equation in radial diretion an be expressed as U U V P u uw v u U W D u r z r r r z r r (7) 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. The right hand side of the equation is the radial pressure gradient term, turbulent fore are P, turbulent fore term, r T ru, T zuw, v T and u r T u, and the diffusion term, D u. The sub-terms in the T. The diffusion term in the equation is small enough and an be ignored omparing with the other terms. The terms of r, r and r at the entral line are alulated by the method as desribed in Appendix. Fore balane in the radial time-averaged Navier-Stokes equation at z. r is shown in Fig. 8(a), whih reveals that the entrifugal fore, pressure gradient and the vertial advetion terms are the signifiant portion of the total balane and the turbulent fore plays little role. Centrifugal fore is the largest term in the orner region, whih inreases with dereasing in radius until reahing to the maximum at around r. 4r and then eventually reahes to at the enter of vortex. The magnitudes of vertial advetion term and the pressure gradient term have a similar tendeny with the entrifugal fore, and the magnitude of the vertial advetion term is larger than that of the pressure gradient term for r. 7r. With an attempt to investigate the turbulent fore in detail, the variations of the sub-terms in the turbulent fore versus radius are illustrated in Fig. 8(b). Two sub-terms, T v and T u, are extremely large in omparison with the other terms and an aute inrease of the magnitudes of and u V v u T in the orner region is observed. The non-zero r.m.s of the flutuating veloities provides that T v and T u reah to infinite at the enter, but they almost anel out with eah other. The reason will be explained in the setion 4.. The tangential veloity an be estimated from the balane between the entrifugal fore, pressure gradient and the vertial advetion terms as follows T v V P A V P V A P U r W r r z (8)

12 Takeshi Ishihara and Zhenqing Liu T ru T v T zuw T u A ru P r A zu T u C r (a) Advetion, pressure gradient, entrifugal fore and turbulene terms r/r (b) Sub-terms of turbulent fore Fig. 8 Radial distribution of normalized radial fore terms at z. r, with (a) representing advetion, pressure gradient, entrifugal fore and turbulene terms, and (b), the sub-terms of turbulent fore r/r.5 V/V.5 Pred. V Cal. V P Cal. V A Cal. V P+A.5.5 r/r Fig. 9 Comparison of simulated and alulated tangential veloities at z. r where, V P and V A are alulated as / P / r r and W U / z r. The alulated tangential veloity based on the model shows good agreement with the simulated one as shown in Fig. 9. This indiates that the inrease of tangential veloity near the surfae omes from not only the pressure gradient term, P r, but also the vertial advetion term A zu. The mehanism of the flow fields an be explained suh that pressure gradient is independent of the height and the ground pressure gradient does not balane with the entrifugal fore beause of the frition. As a result, large radial inflow ours near the ground and auses an inrease in the tangential veloity there.

13 Numerial study on dynamis of a tornado-like vortex with touhing down -A rw T ruw +T uw.5 -A zw.5 T zw P z z/r T w z/r (a) Advetion, pressure gradient and turbulene terms (b) Sub-terms of turbulent fore Fig. Vertial distribution of normalized vertial fore terms at r r, with (a) representing advetion, pressure gradient and turbulene terms, and (b), the sub-terms of turbulent fore.5 Z/r Pred. W Cal. W P Cal. W T Cal. W P+T W/V Fig. Comparison of simulated and alulated vertial veloities at r r The time-averaged Navier-Stokes equation in the vertial diretion an be expressed as W U r W W z P uw w z r z uw r The left hand side onsists of the radial advetion term, A rw, and the vertial advetion term, A zw. The right hand side is the radial pressure gradient term, P z, turbulent fore term, T w, and the diffusion term, D w. The sub-terms in the turbulent fore are T ruw, T zw and T uw. The diffusion term in the equation is small enough and an be ignored ompared with the other terms. uw/ r and uw / r at the entral line are alulated as shown in Appendix. Fig. (a) shows the balane of the vertial fore in the vertial time-averaged Navier-Stokes equation at the enter of vortex. It is observed that the vertial advetion term balanes mainly D w (9)

14 Takeshi Ishihara and Zhenqing Liu with the pressure gradient term and the turbulent fore term. The radial advetion term is zero at all heights in respet of the symmetry of the mean vertial veloity. The vertial advetion term beomes alternately positive and negative with height, and the height of the inversion point in the profile is.5 r. The magnitudes of the pressure gradient and the turbulent fore exhibit a similar tendeny, first inrease then derease with dereasing in height and the maximum values our in the region below the height of. r. Above the height of.4 r, all the terms are almost zero on aount of the relatively stable flow fields in the upper part of the vortex. The sub-terms of turbulent fore at r r are illustrated in Fig. (b). Above the height of.7 r, the sub-terms are about zero. Within the height between.7 r and.4 r the non-zero sub-terms almost anel out with eah other. Near the ground, the sum of uw/ r and uw / r shows negative value, while the sub-term w / z hanges from negative to positive with height. The vertial veloity an be estimated from the balane between vertial advetion term, the pressure gradient term and the turbulent fore term as follow W W W P P W T W T P P P z T S P z T S w w dz dz for for W W () where, the W P and W are alulated as P/ T P S and z Twdz, P S is the pressure at surfae and P is the pressure at the height of z. The sign of W an be deided as follows: W has to be positive at the level very lose to the bottom, sine it is impossible for the partiles to move downward here. With inreasing in elevation, W will be at some height and should be negative further inrease in elevation. The estimated tangential veloity W based on the model shows favorably agreement with the simulated one as shown in Fig.. The vertial veloity, W P, is larger than the simulated result, if the turbulent ontribution is not taken into onsideration. This indiates that the pressure gradient aelerates the vertial veloity while the turbulent fore deelerates that. 4. Dynamis of the tornado-like vortex The Dynamis of tornado-like vortex with touhing down are revealed in this setion. Firstly, the flow pattern of the instantaneous flow fields is examined by the pressure iso-surfae, the vetors as well as the time series of veloities. Then the power spetrums for the flutuating veloities are studied to quantify the organized swirl motion of the tornado-like vortex. Finally, the gust speed of tornado-like vortex with touhing down is alulated to provide valuable information from an engineering point of view. In the following disussions the time is normalized by r / V, whih is the period of the vortex based on the assumption of the solid rotation in the ore where the mean tangential veloity is proportional to the radial distane.

15 Numerial study on dynamis of a tornado-like vortex with touhing down 4. Instantaneous flow fields The three-dimensional pressure iso-surfaes of.7 Pmin are illustrated in Fig., where Pmin is the minimum pressure of the flow fields. The pressure iso-surfaes yield the familiar and lear shape of the tornado. It is apparent that the enter of the vortex is not stationary but assoiated with a swirl motion around the enter of the simulator as shown by the points in Fig.. The vortex reahes to the southern, eastern, northern and western sides at t =33., 36.6, 39.5 and 4. respetively, implying that the angular speed of the swirl motion is muh lower than that of the solid rotation. The swirl motion is found to be organized rather than random or haoti, whih will be larified in the spetrum analysis. At the orner of the vortex, some small eddies appear with fairly rapid evolution and they loate primarily inside the ore of the vortex. Z Z Y Y X X X /r Y/ r -.5 N X/r -.5 Y/ r E W (a) Western side (b) Northern side Z Z Y Y S X X X/r () Southern side r Y/ -.5 X /r Y/ r (d) Eastern side Fig. Evolution of three-dimensional pressure ontour with pressure level of.7 Pmin for one period. Vortex enter moves from () southern side at t =33 to (d) eastern side at t =36.6, (b) northern side at t =39.5 and (a) the western side at t =4

16 Y/r Y/r Y/r Y/r Takeshi Ishihara and Zhenqing Liu Fig. 3 shows the instantaneous veloity vetors at the horizontal ross-setion of z. r. The veloity vetors are not as symmetry as that in Fig. 4(a), and the swirl motion an be learly observed, where the diretion of the wind veloity at the enter hanges periodially due to this swirl motion. When the tornado-like vortex moves to the southern and the northern sides, the veloity vetors are almost parallel to the x axis at the enter, indiating the ourrene of the large radial speed and the small tangential speed, as shown in Figs. 3() and 3(b). On the other hand, when the vortex moves to the eastern and western sides, the veloity vetors are almost perpendiular to the x axis at the enter, whih means a small radial speed and a large tangential speed our, as shown in Figs. 3(d) and 3(a). Hene, the time histories of the radial and the tangential veloities should be periodi. As a result, it an be onluded that the turbulene in the tornado-like vrotex is different from the turbulene in the full developed shear layer and the main soure of the flutuationg horizontal veloities in the tornado-like vortex is the organized swirl motion X/r (a) Western side X/r (b) Northern side X/r () Southern side X/r (d) Eastern side Fig. 3 Vetor fields at the ross setion of z. r for one period. Vortex enter moves from () southern side at t =33 to (d) eastern side at t =36.6, (b) northern side at t =39.5 and (a) the western side at t =4

17 v / V u / V Numerial study on dynamis of a tornado-like vortex with touhing down Sample traes of radial and tangential veloity flutuations from non-dimensional time 3 to 44 at the loation of r r and z. r are shown in Fig. 4. The vertial dashed lines show the time as the tornado-like vortex moves to the southern, eastern, northern and western sides. It is observed that the tangential and radial veloities display signifiant flutuations due to the organized swirl motion. The shapes of the time histories for radial and tangential veloities are similar and periodi. Moreover, the large values of the radial and tangential veloities do not happen simultaneously. When the tornado-like vortex moves to the eastern and western sides, the tangential wind speed is larger than the radial wind speed, on the other hand when the vortex moves to the southern and northern sides, the tangential wind speed is smaller than the radial wind speed, as shown by the points in Fig. 4 indiating a 9 o phase differene between the time histories of the radial and tangential veloities. The sinuous shape for the radial and tangential veloities yields zero time-averaged values and almost the same standard deviations. This is the reason why the sub-terms v / r and u / r anel out with eah other in the radial fore balane at the enter. Fig. 5 shows the time series of the vetor distributions at the ross-setion of y=. The dashed line shows the axis of the simulator. The vertial veloity at the entral line does not hange dramatially at eah elevation, whih an be explained as suh that, even though the organized swirl motion exists, the distane between the enter of the simulator and the enter of the instanteneous vortex does not vary a lot, maintaining.5 r approximately as shown in Fig. 3. The omparison between the time variations of the instantaneous radial and vertial veloity flutuations from non-dimensional time 3 to 44 at the loation of r r and z. r is shown in Fig. 6. The pretty small and random flutuation of the vertial omponent is the indiative and the orrelation between the organized swirl motion and the vertial veloity flutuation is not obvious. It an be onluded that the organized swirl motion is not the main soure of the turbulene for the vertial veloity. As for the effetiveness of turbulent mixing, the vertial veloity is smaller than that aused by the laminar vortex. This is the reason why the turbulent fore dereases the vertial veloity. S E N W tv/r (a) Radial veloity S E N W tv/r (b) Tangential veloity Fig. 4 Simultaneous traes of instantaneous (a) radial veloity and (b) tangential veloity

18 w / V u / V Z/r Z/r Z/r Z/r Takeshi Ishihara and Zhenqing Liu X/r (a) Western side X/r (b) Northern side X/r () Southern side X/r (d) Eastern side Fig. 5 Vetor fields at the ross setion of y= for one period. Vortex enter moves from () southern side at t=33 to (d) eastern side at t=36.6, (b) northern side at t=39.5 and (a) the western side at t=4 S E N W tv/r (a) Radial veloity S E N W tv/r (b) Vertial veloity Fig. 6 Simultaneous traes of instantaneous (a) radial veloity and (b) vertial veloity

19 ns(n)/ w ns(n)/ u ns(n)/ v Numerial study on dynamis of a tornado-like vortex with touhing down - - z=.r z=.5r z=.r z=.r z=.5r z=.r nr/v (a) u omponent nr/v (b) v omponent - z=.r z=.5r z=.r nr/v () w omponent Fig. 7 Normalized spetral densities of veloity flutuations at omponent, () w omponent r r, (a) u omponent and (b) v 4. Spetrum analysis In order to quantify the organized swirl motion of the tornado-like vortex, the power spetra of the veloity flutuations, u, v, w, is alulated by the Maximum Entropy Method (MEM). The normalized spetra of radial, tangential and vertial omponents against a non-dimensional frequeny f nr / V, where n is the natural frequeny in Hz, at r = and r r are illustrated respetively in Fig. 7. The u spetra at radius of r = is shown in Fig. 7(a). At the height of. r the only one peak at about f =.8 is observed whih orresponds to the organized swirl motion and the primary energy of the flutuating radial veloity omes mainly from the swirl motion. At the height of.5 r, there is no apparent peak for the radial omponent, due to the strong mixing of the fluids. Compared with the data at z. r, the primary frequeny at z r shifted slightly toward the high frequeny range and there is another peak at about f =, oresponding to the rotation of the ore of the tornado. The v spetra at radius of r = shown in Fig. 7(b) are similar with those of u omponent and the flutuating energy is onentrated in the low frequeny range. This fat implies that the flutuating tangential and radial veloities ome mainly from the swirl motion. In omparison with the radial and tangential omponents, the spetra of the

20 Takeshi Ishihara and Zhenqing Liu vertial veloity as shown in Fig. 7() are broad, indiating that the flutuating vertial veloity is not organized as the horizontal veloity omponents. Moreover, at high elevation, z r, the flutuating energy is onentrated in the high frequeny range and has a peak at about f =, indiating that the flutuating vertial veloity is mainly attributed to the rotation of the ore of the tornado at high elevations. 4.3 Gust speeds The gust speed is the governing fator in the struture design, whih is often referred to as a "3-seond gust". In this study, the time used to alulate the gust speed is based on the time sale, as shown in Table, where l, v and t denote the length sale, veloity sale and time sale respetively. The information of the tornado in nature is provided by the DOW radar dataset (Dowell et al. 5). Fig. 8 shows the normalized mean speeds, the speed standard deviations and the gust speeds at r r and r. 5r with a height of. r. It is noteworthy to mention that the gust speed is.7 times the maximum mean tangential veloity, V, at the enter, even though the mean speed is zero, indiating that the veloity flutuation by the swirl motion an ause a large gust speed. The mean speed and the gust speed inrease to.4 V and.9v at r. 5r, whereas the standard deviation dereases to.3v, hene the gust speed omes mainly from the mean speed. The high gust speed is the explanation why tornados ause tremendous destrutions near the entral region. Table Sale ratio of tornado in nature to simulated tornado Real tornado Numerial tornado Sale ratio r (m).36 l = 634 V (m/s) v = 7.8 Averaging Time (s) 3.38 t = mean speed standard deviation gust speed Wind Speed / V.5.5 r= r=.5r Fig.8 Mean speed, standard deviation and gust speed at two representative loations

21 Numerial study on dynamis of a tornado-like vortex with touhing down 5. Conlusions The three-dimensional turbulent flow field and dynamis of a tornado-like vortex with touhing down are investigated by using LES turbulene model. The onlusions of this study are summarized as follows: A numerial tornado simulator developed in this study suessfully generates tornado vortex with touhing down and shows a good agreement with those from the laboratory simulator. The tangential veloity reahes to.4 V at r. 5r near the ground and the radial inflow is onentrated in a thin layer near the ground, showing the peak radial inward veloity of about V at r r. The turbulent kineti energy gives a maximum value at the enter of vortex, while the shear stress at the entral line is zero. The inrease of tangential veloity near the ground and the upward-downward vertial veloity at the enter of vortex an be well explained by axisymmetri time-averaged N-S equation. The turbulent fore plays little role in the radial balane, but that ould not be negleted in the vertial balane. The vortex with touhing down is assoiated with a swirl motion, whih is organized instead of random and it is the main soure of the turbulene of the radial and tangential omponents, also the explanation why the turbulent ontribution is negligible in the radial balane. Two typial peaks are observed from the power spetra of the veloity flutuations assoiated with organized swirl motion of the tornado-like vortex. One appeared at about f =.8 at the level lose to the ground orresponds to the organized swirl motion and another peak at about f = orresponds to the rotation of the ore of the tornado. The gust speed at the enter of the vortex is.7 V even though the mean speed is zero, while the gust speed at r. 5r omes mainly from the mean speed. The high gust speed is the explanation why tornados ause tremendous destrutions in the entral region. Referenes Baker, G.L. (98), Boundary layer in a laminar vortex flows, PhD Thesis, Purdue University, West Lafayette, IN USA. Dowell, D.C., Alexander, C.R., Wurman, J.M. and Wiker, J.L. (5), Centrifuging of hydrometeor and debris in tornadoes: radar-refletivity patterns and wind-measurement erros, Mon. Weather Rev., 33, Ferziger, J. and Peri, M. (), Computational method for fluid dynamis, 3rd Ed., Springer. Haan, F.L., Sarkar, P.P. and Gallus, W.A. (8), Design, onstrution and performane of a large tornado simulator for wind engineering appliations, Eng. Strut., 3(4), Howells, P.C., Rotunno, R. and Smith, R.R. (988), A omparative study of atmospheri and laboratory analogue numerial tornado-vortex models, Q. J. Roy. Meteor. So., 4(48), 8-8. Ishihara, T., Oh, S. and Tokuyama, Y. (), Numerial study on flow fields of tornado-like vorties using the LES turbulent model, J. Wind Eng. Ind. Aerod., 99(4), Kuai, L., Haan, F.L., Gallus, W.A. and Sarkar, P.P. (8), CFD simulations of the flow field of a laboratory-simulated tornado for parameter sensitivity studies and omparison with field measurements, Wind Strut., (),

22 Takeshi Ishihara and Zhenqing Liu Lewellen, D.C., Lewellen, W.S. and Sykes, R.I. (997), Large-eddy simulation of a tornado s interation with the surfae, J. Atmos. Si. 54(5), Lewellen, D.C., Lewellen, W.S. and Xia. J. (), The influene of a loal swirl ratio on tornado intensifiation near the surfae, J. Atmos. Si, 57(4), Lewellen, D.C. and Lewellen, W.S. (7), Near-surfae intensifiation of tornado vorties, J. Atmos. Si,, 64(7), Liu, Z. and Ishihara, T. (), Effets of the Swirl Ratio on the Turbulent FlowFields of Tornado-like Vorties by using LES Turbulent Model, Proeedings of the 7th International Colloquium on Bluff Body Aerodynamis and Appliations, CD-ROM. Matsui, M. and Tamura, Y. (9), Influene of swirl ratio and inident flow onditions on generation of tornado-like vortex, Proeedings of the EACWE 5, CD-ROM. Mitsuta, Y. and Monji, N. (984), Development of a laboratory simulator for small sale atmospheri vorties, Nat. Disaster Si., 6, Natarajan, D. and Hangan, H. (), Large eddy simulation of translation and surfae roughness effets on torando-like vorties, J. Wind Eng. Ind. Aerod., 4-6, Nolan, D.S. and Farrell, B.F. (999), The struture and dynamis of tornado-like vorties, J. Atmos. Si., 56(6), Oka, S. and Ishihara, T. (9), Numerial study of aerodynami harateristis of a square prism in a uniform flow, J. Wind Eng. Ind. Aerod., 97(-), Tari, P.H., Gurka, R. and Hangan, H. (), Experimental investigation of tornado-like vortex dynamis with swirl ratio: The mean and turbulent flow fields, J. Wind Eng. Ind. Aerod., 98(), Wan, C.A. and Chang, C.C. (97), Measurement of the veloity field in a simulated tornado-like vortex using a three-dimentional veloity probe, J. Atmos. Si., 9(), 6-7. Ward, N.B. (97), The exploration of ertain features of tornado dynamis using a laboratory model, J. Atmos. Si., 9(6), Wilson, T. and Rotunno, R. (986), Numerial simulation of a laminar end-wall vortex and boundary layer, Phys. Fluids, 9(), CC

23 Numerial study on dynamis of a tornado-like vortex with touhing down Appendix The following introdues the method alulating V / r, v / r and u / r in the radial axisymmetri Navier-Stokes equation and uw/ r and uw / r in vertial axisymmetri Navier-Stokes equation at the entral line. The subsripts r,, x and y are adopted to distinguish the radial and tangential variables in ylindrial oordinate as well as the variables in x and y diretions in Cartesian oordinate. The mean tangential veloity is assumed to be nearly proportional to the radius near the enter of the vortex (see Ishihara et al. ()) and an be expressed as V r (A.) where is a onstant. The entrifuge fore, approahes to at the enter of the vortex. V / r, equals to r near the enter and Table A. Symmetries of variables in Cartesian oordinate u x v y w z u x v y u v z y w z xw Cartesian oordinate S S S S A A *S and A denote symmetry and asymmetry with respet to x and y axes, respetively The turbulent fore for the radial axisymmetri Navier-Stokes equation should equal to that for Cartesian Navier-Stokes equation in x diretion at the enter u r uw v z r u r ux x uxv y y uxw z z (A.) Aording to the symmetry of the Reynolds stresses, as illustrated in Table A, u x / x u v / y should equal to, and uw/ z should equal to u w / z, so the sum of turbulent x y terms, u / r v / r u / r, equals to. The turbulent fore for the vertial axisymmetri Navier-Stokes equation should equal to those for Cartesian Navier-Stokes equation in z diretion at the enter uw uw w r r z uxw x z vyw y z x z wz z and (A.3) w / z should equal to w z / z, uw / r uw/ r an be alulated by u w / x v w / y. x z y z