Budownctwo Archtektura 5 (2009) 29-38 Survey of applcatons of dscrete vortex method n cvl engneerng Tomasz Nowck Lubln Unversty of Technology, Faculty of Cvl Engneerng and Archtecture, Department of Structural Mechancs, e-mal: t.j.nowck@pollub.pl Abstract. Ths paper ncludes an ntroducton to Dscrete Vortex Method (DVM) and ts comparson wth other methods of computer flud dynamcs. In ts second part t focuses on the most mportant applcatons of DVM from the cvl engneerng pont of vew. Some examples known form lterature are presented theoretcal as well as expermental. Key words: Dscrete Vortex Method, Naver-Stokes equaton.. Introducton to Dscrete Vortex Method Dscrete Vortex Method (DVM) s one of numercal methods used for computer smulaton of turbulent flud flows. The orgn of DVM s dated to the 930s. However fast development n vortex methods started n 980s preceded by progress n computer technology. Snce then DVM has been successfully appled to many scentfc and engneerng problems. It has been proved to be useful n flud mechancs, plasma physcs, partcle dynamcs and flud penetraton n porous meda phenomena. For 30 years DVM has been successfully used n cvl engneerng. However t s stll a young method and t s not known wdely among cvl engneers. A poneer computer system based on DVM algorthms s beng developed at the Department of Structural Mechancs of Faculty of Cvl Engneerng wth cooperaton wth Wnd Engneerng Laboratory at Cracow Unversty of Technology... Elementary mathematcal equatons DVM s numercal method developed for solvng the Naver-Stokes equaton (N-S) based on Lagrangan model of partcle tracng. In DVM the N-S equaton s solved by drect smulaton of physcs phenomena. A fnte mesh known from fnte element method or fnte volume method s not used n DVM. Assumng a homogeneous flud wth constant vscosty we can wrte the N-S equaton n the form (): u + ( u ) u = p + u t Re, () where: u velocty feld, p pressure feld, Re Reynolds number, t tme. The equaton () s decomposed by calculaton the rotaton of vector u, whch gves the vortcty transport equaton (2)
30 Tomasz Nowck where: + ( u ) = t Re, (2) = u s a vorcty feld of the flow. The equaton (2) s composed wth two parts: advecton (3) and dffuson (4) part. + ( u ) = 0, (3) t = t Re. (4) The separaton lets us treat a flud flow as two smultaneously phenomena: advecton and dffuson and t s knows as the Splt Algorthm. It s the core dea for computer smulaton of a flud flow n DVM. In DVM a contnuous vortcty feld s replaced wth a dscrete fnte set of vortexes, that s called a vortex cloud. A dscrete vortex s characterzed wth ts strength and poston n a calculaton doman. The strength of a vortex s gven by a formula (5). The equaton (5) can be used for dscretzaton the velocty feld. Γ = L u d r = nds, (5) S where: Γ strength of a vortex, L orented closed curve wth an element dr, S area enclosed by the curve wth an element n. The dscrete vortex set let us reconstruct the velocty feld on the bass of a formula (6). N ux ( ) = Γ Kx ( x ), (6) = where: x = (, xy ) poston n two dmensonal space, K Bot-Savart kernel, N number of dscrete elements. The pressure feld can be calculated wth the Posson equaton (7) p = t u ( u) u. (7) Therefore the vortex cloud completely descrbes the flow feld n an arbtrary pont and tme moment..2. Computer smulaton process Computer smulaton of advecton of a flud flow composed of dscrete elements (Fg. ) conssts n calculatng veloctes for every vortex element accordng to the equaton (6) (Fg. 2) and movng the dscrete elements to new postons on the bass of calculated veloctes and chosen tme step. In order to reduce a numercal error at least a second order procedure should be appled (8).
Survey of applcatons of dscrete vortex method n cvl engneerng 3 n+ 05. n n x = x + u t n+ n n n+ 0. 5 x = x + ( u + u ) t, (8) 2 where: t tme step, a number of a vortex, n, n+ number of a smulaton step, n+0,5 ntermedate smulaton step Fg.. Dscrete vortex set. Fg. 2. Resultant velocty determned for a chosen vortex partcle Fg. 3. Translaton of vortex partcles. Computer smulaton of dffuson of a flud can be realzed n a few alternatve ways. The most proper for flows characterzed wth hgh Reynolds Number t the Brownan moton methods that results form the soluton of equaton (4). In the dffuson step of computer smulaton every vortex element s moved from ts poston accordng to formula (9):
32 Tomasz Nowck x n+ = x n + p( ) = e 4πν t 2 4ν t, (9) where: ν knematc vscosty; dsplacement whch module s a Gaussan random varable and angle a random varable wth a unform dstrbuton. Durng smulaton process boundary condton must be fulflled: no-throughflow (0) and no-slp () (Fg. 4). Fg. 4. Boundary condtons. u n = ( u + r ) n, (0) A A b b b b A A u s = ( u + r ) s, () where: u velocty of the flud flow, u b velocty of a body, angular velocty of the body, n A, s A unt vectors n pont A. The no-flow-through condtons says that the flud does not penetrate the body whle the no-slp condton sgnfes that the flud stcks to the body s surface. Despte ther smplcty realzaton of the boundary condton pose the key problem n DVM. It results from the fact that the condtons are formulated for the velocty feld but they must be realzed n terms of dscrete vortcty..3. Dscrete Vortex Method n comparson wth other methods of computatonal flud dynamcs Dscrete Vortex Method s an alternatve to mesh based methods used for solvng Naver-Stokes equaton. If the most mportant methods such as Fnte Dfference Method (FDM), Fnte Element Method (FEM) and Fnte Volume Method (FVM) are based on an arbtrary chosen mesh and requre addng extra smplfed models of turbulence such as Reynolds Average Naver-Stokes (RANS) or Large Eddy Smulaton (LES). The DVM does not make use of a calculaton mesh and smulates the phenomenon of turbulence n drect way. The lack of mesh makes the method very attractve for computer smulaton of turbulent flows because the vortex elements are free to propagate n a computatonal doman. The freedom of
Survey of applcatons of dscrete vortex method n cvl engneerng 33 propagaton enables the vortexes to shape the phenomenon of turbulence durng drect smulaton process. The attractveness of the Dscrete Vortex Method s balanced wth ts computatonal cost whch s much hgher n comparson to FDM, FEM or FVM. For n vortex elements mutual nteractons between them entals the need to perform n 2 calculatons of veloctes for one smulaton step, whch s descrbed n lterature as the n-body problem. For a typcal engneerng task there s a need to use about 5 0 4 elements and perform on them approxmately 0 5 calculaton steps, whch makes n total (5 0 4 ) 2 05 = 0 3 sngle numercal calculatons. The estmated computaton cost s too hgh for contemporary computer hardware. The soluton for the stuaton s usng fast n-body smulators. The smulators dvde the smulaton doman nto separated regons. Interactons wthn one cell are calculated n drect way but the longer dstance nteractons are calculated from a group to a group rather then from a vortex to a vortex. The avodance of drect nteractons works n reducton of calculatons complexty from O(n 2 ) to O(n ln(n)), whch results n reducton smulaton tmes form years to hours. 2. Survey of applcaton of Dscrete Vortex Method n cvl engneerng 2.. Parameters of flow over a buldng The cted example concerns a real buldng ncluded n Texas Tech Unversty campus [2]. The analyzed object was cubcod n shape of dmensons 9. m - 3.5 m n plane and 4 m n heght. The buldng was equpped wth measurement utltes allowng montorng parameters of wnd flow over t. It was localzed n a flat open area far from other buldngs. A power law boundary-layer velocty wnd profle wth an exponent a = 0.8 and a mean speed at the buldng heght 8.6 m/sec was assumed. The wnd flow had an angle-of-attack 90 wth the reference to the longer wall of the object. The problem was smplfed to a two dmensonal flow case and aeroelastc effects were omtted. The numercal smulaton was performed for the Reynold s number Re = 2.3 0 6. The number of vortex partcles vares form 2 000 to 24 000 durng the computatonal process. Numercal results were compared wth expermental data. Selected results have been presented at fgures form 5 to 7. Fgures 5 and 6 concerns the same moment of tme. Fg. 5. Streamlnes of the flow over Texas Tech buldng (vsualzaton) [2].
34 Tomasz Nowck Fg. 6. Pressure feld over Texas Tech buldng (vsualzaton) [2]. Fg. 7. Mean pressure coeffcents on Texas Tech Buldng s walls (dmensonless) [2]. The quoted example shows good conformty between numercal smulaton results and expermental data. The most mportant achevement s the shape of the lne obtaned from computer smulatons. It shows that DVM algorthms are able to model the phenomena of turbulence that occurs near edges of a body lke the place where wall connect wth the roof. The dscrepancy between experment and calculatons, that can be notced n the Fg. 7, comes from the fact that the buldng was not long enough (the length to depth rato was equal.5) to treat the flow as two dmensonal and the nfluences of walls parallel to the flow should not have been neglected and three dmensonal analyss of the flow would have been more proper n ths case. However, the dfferences are small and do not dsqualfy the results wth a cvl engneer s pont of vew. Researches presented n [2] show a wde area of applcatons n cvl engneerng practce. It s possble to estmate wnd loads on buldng. Both the statc mean and dynamc values can be obtaned. The results are of hgh mportance for a structural engneerng whose role s to ensure safety of a buldng structure. Except forces actng on a buldng s planes t s easy n DVM to obtan physcal parameters of flow n an arbtrary pont wthn a smulaton doman. It s possble to follow changes of pressure or velocty of flud at a chosen pont n space. Therefore trouble spots such as ar vents or outlets or chmneys can be tested to possblty of reverse thrusts. Vsualzaton of ar flow (Fg. 5) enables to ndcate areas wth deterorated wnd comfort. All of the mentoned analyss can be made at the desgn stage as well as for an exstng buldngs. Moreover complcated geometry of an object s not an obstacle for DVM, because t does not make use of calculaton mesh.
Survey of applcatons of dscrete vortex method n cvl engneerng 35 2.2. Determnaton of aerodynamc characterstcs Researches presented n [3] concern determnaton coeffcents of aerodynamc resstance: drag C X, lft C Y and moment C M. Square and rectangular sectons (Fg. 8) were analyzed aganst wnd acton wth a dfferent angle of ncdence. The shape rato L/B vares from /4 to 3 and the angle of ncdence a from 45 to 90. Two dmensonal flow feld was assumed. Durng the analyss Reynolds number came to 2 0 4. Fg. 8. The shape beng analyzed. Fgures from 9 to present calculaton results confronted aganst expermental data for the case of L/B =. Fg. 9. Mean drag coeffcent vs. angle of ncdence (dmensonless) [3]. Fg. 0. Mean lft coeffcent vs. angle of ncdence (dmensonless) [3].
36 Tomasz Nowck Fg.. Mean moment coeffcent vs. angle of ncdence (dmensonless) [3]. Quoted results obtaned from DVM analyss may confrm n the convcton that the method can be successfully appled to determne coeffcents of aerodynamc resstant. The characterstc are needed for cvl engneers whle desgnng slender structures for whch wnd acton s the most mportant load. It concerns cable stayed masts, towers, suspended or cable stayed brdges. The coeffcents are evaluated expermentally n the long and expensve process. Any changes n a project of a structure nvolves repeatng of the process. DVM algorthms enables to shorten the tme of desgnng and reduce costs. 2.3. Predcton of dynamc response of a structure wth aeroelastcty effects Undenable superorty to other method to computatonal flud dynamc DVM presents n analyss ncludng aeroelastc nteracton. An example of such smulaton one can fnd n [4], where the authors presents ther results of dynamc response of a brdge glder placed n a turbulent wnd flow. They used DVM to examne aeroelastc stablty of Tacoma Narrow Brdge and compared the results wth hstorcal expermental data. Tacoma Narrow Brdge collapsed 7 th November 940 as a result of ntense flutter caused by wnd blowng wth mean speed 9 m/sec. Accordng to hstorcal resources the angle of brdge deck reached value of 30 just before the catastrophe. Numercal smulatons presented n [4] concerned only a deck of the brdge (Fg. 2). Because flutter occurs n center area of a brdge and brdge spans are at least a few dozen tmes longer then ther wdth lmtaton smplfcaton to two dmensonal analyss are justfed and does not ntroduce bg errors. Fg. 2. Dmensons of Tacoma Narrows Brdge s grder. Computer calculatons reconstructed the phenomena that happened n 940. Numercal experments conducted for wnd speed n a range form 5 m/sec to 20 m/sec.
Survey of applcatons of dscrete vortex method n cvl engneerng 37 They confrmed that the crtcal speed for flutter was 9 m/sec. Vsualzaton of the results showed that the ampltude of perodc torsons came to 30 (Fg. 3). Vertcal dsplacement are presented n Fg. 4. Fg. 3. Torsonal flatter of the grder [4]. 0.02 0.0 0.00-0.0-0.02 0 5 0 5 20 25 30 35 40 45 50 tu/b Fg. 4. Dmensonless vertcal dsplacement of the grder vs. dmensonless tme [4], where h dsplacement of o brdge, B wdth of the deck, t tme, U wnd speed. The convergence between numercal results and known from hstory catastrophe encourage to state that DVM analyss should always be nvolved n the process of desgnng suspended or cable stayed brdges. It enables to dscover and elmnate the most prone to aeroelastc nstablty structure varants. Snce the cost of computer smulatons s much more lower then the cost of expermental test t s possble to examne more varants n shorter tme, choose the best one and test t n a wnd tunnel laboratory. The quoted example concerns a brdge structure but the aeroelastc analyss can be performed to any knd of buldng structure such as masts, towers, sngle cables, etc. It s worth mentonng that aeroelastc smulaton are very dffcult to carred out usng mesh based methods (FEM, FVM) because a movement of a body n a flow requre remeshng at every calculaton step.
38 Tomasz Nowck 3. Summary Takng nto consderaton fast development n computer technology the Dscrete Vortex Method can be treated as a very promsng and versatle tool of wnd engneerng. Computer programs based on DVM algorthms have proofed to be useful n a process of desgnng structures for whch wnd dynamc load s the most mportant one. Computer programs based on DVM algorthms are sure to become a regular engneerng tool n the same way computer programs based on FEM or FVM are used nowadays. They wll be used to estmate wnd load on buldngs, ther dynamc response to wnd and for wnd comfort assessment. However, one must state that DVM s a relatvely young calculatng method and stll requres ntensve scentfc researches before t become a regular desgn engneerng tool. At present t can be consdered a valuable ad for tradtonal methods of wnd engneerng. References [] Cottet G.H., Koumoutsakos P.D., Vortex Method: Theory and Practce, Cambrdge Unversty Press, 2000. [2] Turkyyah G., Reed D., Yang J., Fast vortex methods for predctng wnd nduced pressures on buldngs, Journal of Wnd Engneerng and Industral Aerodynamcs 58, 995. [3] Taylor I., Vezza M., Predcton of unsteady flow around square and rectangular secton cylnders usng a dscrete vortex method, Journal of Wnd Engneerng and Industral Aerodynamcs 82, 999. [4] Larsen A., Walther J.H., Aeroelastc analyss of brdge grder sectons based on dscrete vortex smulaton, Journal of Wnd Engneerng and Industral Aerodynamcs 67&68, 997.