FLAT PLATE BOUNDARY LAYER SIMULATION USING THE VORTEX METHOD

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1 Proceedngs of COBEM 7 Copyrght 7 by ABCM 19th Internatonal Congress of Mechancal Engneerng November 5-9, 7, Brasíla, DF FLAT PLATE BOUNDARY LAYER SIMULATION USING THE VORTEX METHOD Vctor Santoro Santago vsantoro@g.com.br Gustavo C. R. Bodsten Unversdade Federal do Ro de Janero, Departamento de Engenhara Mecânca Pol/COPPE Centro de Tecnologa, Bloco G, sala 4 Ilha do Fundão, Ro de Janero, RJ Brasl gustavo@mecanca.coppe.ufr.br Abstract. The Method has been used to smulate a large number of external flows around bodes. Despte the tremendous mprovement acheved durng the last decades, ths powerful mesh-free technque requres specal attenton to model the vortcty generaton process, the lagrangan transport of vortcty by dffuson, and the hgh computatonal cost of the lagrangan transport of vortcty by convecton. Ths work proposes a new and effcent twodmensonal algorthm that mproves the vortcty generaton and the dffuson models. In addton, an Adaptve Multpole Expanson code s employed to calculate the nduced veloctes of the vortces, whch promotes an enormous decrease n the convectve computatonal effort. Lamb vortces are generated near the wall by dffuson, and the Corrected Core-Spreadng Method s used to smulate the vortcty dffuson n the boundary layer and wake. The convectve moton of the vortex cloud s calculated usng the Adams-Bashforth second-order tme-marchng scheme. The algorthm s tested for the well-known problem of two-dmensonal, ncompressble flow over a flat plate. The agreement wth the flat-plate Blasus soluton ndcates that the algorthm furnshes a good representaton of the generaton, dffuson and convecton processes of vortcty n the flow. Keywords: method, corrected core-spreadng method, adaptve fast multpole method, flat-plate boundary layer, Blasus boundary layer 1. INTRODUCTION Ths paper descrbes a smulaton of the flow over a flat plate usng the Method. Some mprovements on the vortex method proposed by Santago and Bodsten (6) are here mplemented and appled to the well-known Blasus boundary layer flow problem. Ths benchmark problem s chosen because of the exstence of an exact soluton that can be used for comparson. The flat plate boundary layer flow problem s perfect for the evaluaton of the convectvedffusve vortcty transport mechansms that are carred out at durng the smulaton. In addton, we pay specal attenton to the vortcty creaton mechansm at the wall and ts subsequent sheddng nto the flow feld. Ths step of the algorthm s responsble for the transfer of the exact amount of the vortcty generated at the wall nto the flud by dffuson and also for the determnaton of the adequate poston to place the nascent vortces wthn the flud, n the neghborhood of the plate. The smulaton of the Blasus flat plate problem allows for a detaled assessment of the new numercal schemes that have been developed wthn the vortex method framework to smulate hgh Reynolds number flows. The basc dea of Method reles on the dscretzaton of the vortcty feld nto a cloud of free vortex blobs to smulate the convectve-dffusve transport of vortcty n the flow. Dscrete vortex blobs are generated n the neghborhood of the sold wall n order to satsfy the no-slp and the no-penetraton boundary condtons, and they move n a Lagrangan manner to satsfy the vortcty transport equaton. In ths process, advantage s taken of the property that the boundary condton at nfnty s automatcally satsfed due to the decay of the bass functons used to model the vortcty feld of these vortces. In order to determne the evoluton of the flow fled, these vortex blobs move accordng to vortcty-velocty relatons. The Lamb vortex model s employed to avod the sngular behavor of pont vortces at the orgn, whch ntroduces a core for each dscrete vortex. The dffuson process of vortcty s smulated employng the Corrected Core-Spreadng Method (Ross, 1996). In ths methodology, the vortex core radus grows n tme from a mnmum value up to a maxmum pre-determned value. At ths pont, the vortex splts nto four new vortces wth smaller core radus and strength, replacng the orgnal vortex. The newly born vortces are postoned n the neghborhood of the orgnal vortex, such that they mantan overlappng of ther core radus. Ths scheme s accurate and converges fast, as long as core overlappng s mantaned at all tmes. On the other hand, ths procedure mposes an outstandng computatonal effort because the number of vortces n the cloud grows exponentally n tme. To deal wth ths problem, the mergng and cut-off algorthms proposed by Ross (1997) are mplemented. The frst algorthm merges vortces that have ther radus n excessve overlappng, but leavng suffcent overlappng needed to ensure convergence of the Method. The second algorthm provdes a lower bound for the value of the vortex strength that can be reached by the vortces, preventng vortces from splttng durng the dffuson process when ther crculaton become smaller than a specfed value.

2 A hgh computatonal cost s the man problem of the convectve vortcty transport modeled by the Lagrangan vortex moton, snce the velocty feld nduced by the vortces on each other has to be evaluated at the postons of all vortces. Ths vortex-vortex nteracton has an operaton count of the order of Nv, where Nv s the total number of vortces. For ths reason, the Adaptve Fast Multpole Method, developed by Carrer et al. (1988), s used to accelerate the nduced velocty calculaton. Ths scheme reduces the operaton count down to the order of N v. Fgure 1 shows a schematc dagram of the numercal algorthm that we have mplemented. The flat plate dscretzaton step s responsble for settng up N elements on the plate and for determnng the coordnates of the center of each element, called control ponts, where the boundary condtons are mposed. Next, the crculaton of each new vortex blob to be created s determned. The postons n the neghborhood of the plate where these new vortex blobs are placed are then calculated such that the no-slp boundary condton s satsfed. We show that ths poston has great nfluence on the results of the smulaton, snce t contrbutes to mpose a zero-velocty value at the wall when evaluatng the velocty feld. Vortces that cross the plate durng ether the convectve step or the vortex splttng process of the dffuson step are reflected back nto the flow and placed at the correspondng symmetrc poston. Mergng of vortces wth excessve overlappng and cut-off of the splttng process when the vortex strength decreases below a specfed value are numercal technques employed to lmt the computatonal effort. Velocty Evaluaton Flat plate dscretzaton No-slp condton Creaton Dfuson Reflecton Mergng cut-off Convecton End Reflecton Poston Update Fgure1. Smplfed schematc dagram of the numercal algorthm.. MATHEMATICAL FORMULATION The flat plate boundary layer flow develops from an mpulsve start of a unform flow U over a two-dmensonal flat plate algned wth the flow and dscretzed nto N elements, where the no-slp and the no-penetraton boundary condtons are satsfed at the control ponts. The nduced velocty feld at each control pont s the result of the superposton of the unform potental flow U and the cloud of free vortces present n the flow. The reflecton method (Lews, 1991) s appled to ensure that the no-penetraton boundary condton s satsfed. In other words, f M vortces are generated above the plate, then M mage vortces wth opposte strength are also generated below the plate at the correspondng mrror-mage poston. The vortex blobs have a vortcty bass functon that s modeled usng the Lamb vortex model, and the vortctyvelocty relaton for the nduced velocty feld q may be wrtten as Γ r q ( rt, ) = 1 exp, (1) πr σ where Γ s the vortex crculaton (or strength), σ s the core radus that desngularzes the blob and r s the dstance from the vortex locaton to the pont where the nduced velocty s calculated..1. No-slp boundary condton The velocty component n the x-drecton evaluated at the th -control-pont of the plate due to the unform ncdent flow wth freestream speed U and all M free vortex blobs n the cloud and ther mages s gven by M ΔΓ y r = 1 r σ. () 1 γ = U - 1-exp - π Equaton () determnes the vortcty per unt length γ(s ) of each plate element of length Δs to be shed nto the flow n the form of L nascent vortces wth crculaton γ(s )ΔS /L. Ths amount of vortcty dstrbuted over the L new vortces s created n the flow to cancel exactly the velocty at the plate and, therefore, to satsfy the no-slp condton.

3 Proceedngs of COBEM 7 Copyrght 7 by ABCM 19th Internatonal Congress of Mechancal Engneerng November 5-9, 7, Brasíla, DF.. creaton After evaluatng the vortcty γ from Eq. (), the poston of new vortces s establshed such that the no-slp condton s satsfed. It s desrable to choose the poston of the center of the newly created vortex and ts dsplacement ε from the flat plate n such a way that the no-slp condton contnues to be vald at the begnnng of next step. Ths s an mprovement when compared to the classcal procedure that sets ε = σ, but the problem leads to an unusual nonlnear system of algebrac equatons. It s possble to employ an alternatve procedure to calculate ε for control-pont such that the no-slp boundary condton remans vald. To avod the non-lnear system of equaton, we approxmate all ε,, by ther values at the prevous tme step. The unknown ε s then calculated explctly and an teratve procedure s mplemented for all ε,, untl convergence to a specfed value s reached. Fgure shows the vortex poston durng the vortex creaton process. ε element ε + 1 ε + 1 element +1 ε + element + Fgure. Creaton of L vortces for all N control ponts. The approxmaton for ε,, consderng the nfluence of all new M = LN vortces, ncludng ther mages, leads to the followng equaton LN t-1 γ γ ΔS γ ΔS ε =, (3) πlε πlr = 1 t-1 where Eq. (3) s solved for ε ; the value of ε s obtaned from the prevous tme step and γ s calculated from Eq. (). Fgure 3, generated usng the results from smulatons wth and wthout the use of the model above, shows how the use of Eq. (3) can mprove the evaluaton of velocty feld near the wall. The smulaton obtaned by settng ε = σ s very poor compared to the smulaton that evaluates ε from Eq. (3). Ths new model s more accurate to produce both good force calculaton on the surface and good development of the smulaton as a whole. Blasus soluton ε =σ ε from Eq. (3) Fgure 3. Comparson between the velocty feld near the wall generated wth ε =σ and ε calculated for Eq. (3).

4 The dffuson layer that develops durng the vortcty generaton process at the wall, shown n Fg. 4, has an area equal to the product of the Raylegh dffuson dsplacement thckness, δ =,76 Δ tre, to the length of element, ΔS. Ths dffuson area and the number of vortces L created per plate element are used to calculate the radus σ of the nascent vortces accordng to v ΔS δ σ = v. (4) π L The number of vortces L created per plate element s the nteger number obtaned from L = INT(ΔS /δ v ) +1, where ΔS /δ v s the aspect rato of the dffuson layer. Snce ΔS and δ v are kept constant durng the smulaton, L s the same for all plate elements durng all tme steps. The crculaton of each new (nascent) vortex s calculated accordng to γ ΔS Γ k =, 1 k. L. (5) L ΔS δ v.3. Feld evaluaton control-pont Fgure 4. The dffuson area orgnatng from the vortcty creaton at plate element. The velocty feld s evaluated usng the vortcty-velocty relaton obtaned from the Bot-Savart law and the Lamb vortex model descrbed by Eq. (1). Because we use the reflecton method (Lews, 1991) to mpose the no-penetraton boundary condton on the plate, our formulaton ncludes the mage vortces as well. Therefore, the x and y components of the velocty vector q nduced by the th -vortex at the th -doman pont, summed over all the N v vortces n the cloud and ther mages, are gven by N v ΔΓ y - y r 1 y + y r Δ u = 1- exp -5, exp -5, 57, = 1 π r1 σ r σ (6a) N v ΔΓ x -x - r x x r 1 Δ v = 1- exp -5, exp -5, 57, = 1 π r σ r1 σ (6b) = ( ) + ( ) r ( x - x) ( y y) where r1 x -x y - y, the dstance from the vortex core center s Eqs. (6) s replaced by the followng equatons = + + and σ = rmax. The tangental velocty s a maxmum when rmax. When the th -pont s the th -vortex, the frst term n curly brackets n ΔΓ y Δ u = - 1- exp -5, 57 4π y, (7a) σ Δ =. (7b) v It s necessary to correct the vortcty feld when the vortex core crosses the flat plate, as shown n Fg. 5, because part of the vortcty s now nsde the wall and t does not contrbute to the vortcty and velocty felds. Ths correcton s mplemented by modfyng the vortex crculaton accordng to θ θ -θ1 -θ1 φ ( π ) * Γ r * Γ= ωda= exp = 1 Γ rdrdθ, (8) πσ σ where θ 1, θ and φ are shown n Fg. 5. Consequently, the vortex core radus s also corrected by the equaton

5 Proceedngs of COBEM 7 Copyrght 7 by ABCM 19th Internatonal Congress of Mechancal Engneerng November 5-9, 7, Brasíla, DF ( φ senφ) * σ = 1 σ. (9) π Equaton (9) s derved from the relaton ** plate and A s the area below the flat plate. * ** A = A T - A, where A T the total area of the vortex, A * s the area above the flat y q s f q 1 x Flat plate Fgure 5. core radus crossng the flat plate..3. dffuson In the vortex method the dffusve transport of vortcty s related drectly to how the vortcty created at the wall s shed and dffused nto the flow. The frst paper to attack the problem of vortcty dffuson n Lagrangan computatonal flud dynamcs smulatons s due to Chorn (1973), who proposes a stochastc method, called the Random Walk Method, based on random dsplacements of the vortces. Ths method was consdered for some tme the only way to deal wth vortcty dffuson n a Lagrangan descrpton of the flow. In the last couple of decades, many determnstc models have been developed (Santago and Bodsten, 6). One of the most popular s the orgnal Core-Spreadng Method (Leonard, 198), whch allows the vortex core radus to grow n tme accordng to a dffuson rate that s a soluton of the vortcty dffuson equaton. Ths method was proved by Greengard (1985) to dverge from the Naver- Stokes equaton. The Corrected Core-Spreadng Method (CCSM), due to Ross (1996), proposes the splttng of the vortex nto four new vortces as soon as the radus reaches a maxmum value. Ths modfcaton ntroduced by Ross has turned the method convergent and, therefore, t has become agan a usable dffuson technque. Ths method provdes accurate determnstc results and t s very smple to mplement numercally. The basc dea of CCSM s to let the vortex core radus grow n tme from a mnmum value,σ mn, to a maxmum value, σ max, smulatng the vortex dffuson process. Ths growth s controlled by a numercal parameter α, whch has values n the range [,1], where σ max s nput to the algorthm and σ mn = ασmax. The splttng process occurs when the vortex reaches a radus greater than σ max. So every tme σ becomes greater than σmax, the code splts the vortex n four new vortces wth crculaton Γ/4, unformly placed at a dstance r from the orgnal vortex calculated to conserve the vortcty second-order moment. The radus growth rate and the dstance r are gven by σ t = ν, (1) r σ α = 1, (11) where ν s the knematc vscosty..4. Reflecton As a result of the refnement of the vortcty feld due to dffuson and of the tme dscretzaton of the convectve step, a vortex may end up at a poston below the flat plate. Ths may occur after the dffuson step, when a new vortex s created below the plate from the splttng process, or after the vortex convecton, when the vortex dsplacement calculated wth the Adams-Bashforth second-order tme-marchng scheme places the vortex below the wall. Hence, t s necessary to call the reflecton subroutne after these both steps, as t can be seen n the Fg. 1.

6 .5. Mergng and cut-off Compared wth the Random Walk Method (Chorn, 1973), CCSM ncreases the accuracy because t s a determnstc method that dscretzes the vortcty feld by splttng the vortces n the cloud. However, ths characterstc ncreases the computatonal cost of CCSM unboundedly, snce the number of vortces ncreases exponentally. Ths problem can be dealt wth usng Ross s (1997) mergng and a cut-off schemes. Mergng vortces s the acton that substtutes a set of overlappng vortces by one only. Ths procedure approxmates the feld generated by ths vortex but t keeps the approxmaton error under control. In ths paper, we use the algorthm proposed by Ross (1997), where the vortcty feld s approxmated by a lnear combnaton of N basc functons, as follows ω ( x) N Γ x- x = exp - = 1 4 4, (1) πσ σ where each basc functon s defned by three parameters (Γ, x, σ ), correspondng to the vortcty, the poston and radus of the vortex, respectvely. A maxmum and mnmum dscretzaton resoluton must be establshed to start the algorthm. Ths s done choosng l m σ l M to obtan good vortex radus overlappng and avod regons wth nsuffcent dscretzaton. In our algorthm, we set l M = σ max and lm = σ mn. After mergng n vortces, the mergng error s Γ e( x) = exp exp 4πσ 4σ σ 4σ x n Γ x σ x = 1 γ. (13) The substtuton of a set of vortces by one vortex s carred out such that the zeroth, frst and second moments of vortcty are mantaned accordng to the followng equatons Γ = n =1 Γ, Γ x = Γ x and n =1 n 4Γ σ = Γ 4 + σ x x. (14a, b, c) = 1 Ross s (1997) algorthm s wrtten to dentfy a set of n vortces accordng to Eqs. (14a, b, c), boundng the error by Eq. (13). Wth ths methodology, the error s bounded n space and the computatonal vortcty feld only experences controllably small nstantaneous perturbatons. Thus, the largest error n the entre smulaton s smlar to the largest local error. The cut-off scheme s another way to control the problem sze,.e., the total number of vortces n the cloud. The cut-off scheme nterrupts the splttng process of vortces wth suffcently small crculaton. Ths s a mechansm that lmts the dscretzaton of the vortcty feld and helps to prevent the exponental growth of the number of vortces. More detals can be seen n Ross (1997). In the smulaton presented n ths work the cut-off value was chosen to be convecton The drect evaluaton of the convectve velocty of all N v vortces s a par-wse vortex-vortex nteracton, whch has computatonal cost of the order of Nv. Alternatvely to the drect vortex-vortex calculaton, ths work uses the Adaptve Fast Multpole Method (AFMM), developed by Carrer et al. (1988), to calculate the velocty of each vortex. Ths acceleraton procedure has the ablty to reduce the computatonal effort down to order N v, dvdng the vortex cloud nto boxes that are consdered far enough apart by a numercal parameter. The velocty nduced by one box at the others s calculated expandng the Bot Savart law nto a Laurent seres. The number of terms n the Laurent seres expanson s determned for a desred accuracy, and the doman s dvded nto smaller and smaller boxes untl each box contan a maxmum number of vortces per box; at ths pont, the spatal refnement stops. The nteractons are carred out from one box to all other boxes, and vortces nsde the same box are subect to drect vortex-vortex calculaton. Santago et al. (6) have performed a seres of tests wth the algorthm of the AFMM and observed that, when the vortex densty n the wake becomes very hgh, the smallest boxes become of the order of the vortex core radus. In ths case, the algorthm loses performance nstantaneously and the maxmum number of vortces per box must be ncreased. As soon as the maxmum number of vortces per box s ncreased, the algorthm performance returns to ts orgnal computatonal cost of order N v.

7 Proceedngs of COBEM 7 Copyrght 7 by ABCM 19th Internatonal Congress of Mechancal Engneerng November 5-9, 7, Brasíla, DF Another partcular aspect of the AFMM of Carrer s et al. (1988) s that the near approach dstance for desngularzaton can be mplemented to work together wth the man routne. Ths s equvalent to sayng that the algorthm can use the Lamb vortex model, but the sze of the smallest box has to be greater than the radus of the vortces, because all vortces nsde the same box has to undergo drect calculaton, as explaned above. The Adaptve Fast Multpole Algorthm s a very useful tool that allows longer smulatons to be executed wth better dscretzaton of the vortcty feld, snce t makes possble to work wth larger wakes (larger problem szes). Too bg problem szes are prohbtve f usng the drect calculaton method based on the Bot-Savart law. Because we use the reflecton method to mpose the no-penetraton boundary condton, t becomes necessary to nput the mage cloud nto the AFMM routne together wth the physcal vortex cloud. Although the convectve step works wth N v vortces, t s stll advantageous, snce the reducton n the CPU tme offered by Nv computatonal cost overcomes the ncrease n the problem sze..7. Poston Update The step that updates the poston of each vortex n the cloud s also responsble for the determnaton of ther total dsplacement. Ths calculaton s performed at the end of tme step because t s necessary to know beforehand the convectve velocty calculated wth the algorthm of the AFMM. Knowng the velocty of all vortces n the cloud, we employ the Adams-Bashforth second-order tme-marchng scheme to update the poston of the vortex cloud, whch requres the nformaton of the velocty feld calculated at the current and the prevous tme steps. So, we use the frstorder Euler scheme n the frst step of the smulaton and for all the nascent vortces, and a second-order Adams- Bashforth scheme for the remanng vortces of the cloud at any tme step. The x and y components of the dsplacements calculated by the frst-order Euler scheme and the second-order Adams-Bashforth scheme, respectvely, are gven by Δ x = ut () Δ t and Δ y = v() t Δt, (15a, b) C C Δ x = [ 1,5 ut ( )-,5 ut ( - Δt) ] Δ t [ 1,5()-,5(- )] C 3. RESULTS AND DISCUSSION and Δ y = vt vt Δt Δt. (16,a, b) C A good smulaton usng the vortex method s essentally related to a good representaton of the vortcty feld everywhere n the rotatonal flow regon. In partcular, the vortcty and ts flux at the sold surface calculated to satsfy the boundary condtons are very mportant to determne the entre flow feld and the surface forces. The models proposed n ths paper generate a velocty feld n good agreement wth the Blasus soluton for the boundary layer flow over a flat plate, as one can see n the Fgs. 6. These fgures show converged soluton at tme t = 1.5 for the u-velocty profle as a functon of η yu ( / νx), evaluated at three dfferent x statons and compared to the Blasus soluton, for Re = 1, Δ t =.1, α =.9, σ =.1. The velocty profle present better agreement as x approaches the tralng edge of the plate, as expected. Our results also present better agreement to the Blasus soluton when compared to the numercal soluton obtaned by Lews (1999) for Re = 5, as shown n Fg. 7. The ntal convergence evoluton of the smulaton as tme progresses can be seen n Fg. 8. (a) x =.5; (b) x =.5; (c) x =.75; Fgure 6. Velocty profles at three x statons along the plate; for Re = 1, Δ t =.1, α =.9, σ =.1.

8 Fgure 7. The u-velocty profle at x =,5 obtaned by Lews (1991) for Re = 5. Fgure 8. Tme convergence for Re =1. The wake represented by the cloud of vortces at t =.5 s shown n Fg. 9, for Re = 1. Ths fgure llustrates the growth of the boundary layer along the plate and the hgh vortcty resoluton obtaned wth our vortex method. Our smulaton ends up wth N = 59,837 vortces at t =.5. v Fgure 9. Poston of the vortces n the wake at t =.5 for Re =1.

9 Proceedngs of COBEM 7 Copyrght 7 by ABCM 19th Internatonal Congress of Mechancal Engneerng November 5-9, 7, Brasíla, DF 4. CONCLUSIONS Ths work ntroduces several modfcatons and mprovements on prevous models for the vortex method to study the complex physcal phenomenon assocated wth convectve-dffusve problems such as the boundary layer development over a flat plate. One of the man mprovements s the use of the multpole expanson technque through the use of the AFMM, whch allows hgh-resoluton smulatons to be carred out for longer tmes. The mplementaton of ths method was necessary n order to replace the stochastc Random Walk Method by the determnstc and more accurate Corrected Core-Spreadng Method (Ross, 1996), whch ncreases the number of vortces n the cloud every tme step. The CCSM has proved to gve better representaton of the vortcty feld. Fnally, the new vortex generaton scheme descrbed n ths paper adds an mportant correcton procedure to calculate the vortcty and velocty felds near sold boundares. As the results for the Blasus flat plate smulaton show, the vortex method presented here s very encouragng to descrbe n detal the hgh Reynolds-number flows around bodes. Regardless of the good results shown here, work stll needs to be done. The slope of the velocty feld near the wall stll requres some mprovement to be obtaned. Ths nformaton s key to determne the skn frcton factor and, consequently, the frcton force on the surface of the plate. 5. ACKNOWLEDGEMENTS The authors would lke to acknowledge Prof. Lesle Greengard for allowng us to use hs orgnal FORTRAN code of the Adaptve Fast Multpole Method, whch has been slghtly modfed by the authors to be used n ths work. The authors would also lke to acknowledge the Brazlan Army and CNPq, through grants no. 346/4-5 and 47795/4-3, for the fnancal support of ths research proect. 6. REFERENCES Carrer, J., Greengard, L. and Rokhln, V., 1988, A Fast Adaptve Multpole Algorthm for Partcle Smulaton, SIAM Journal of Scentfc Statstcs and Computaton, Vol. 9, n. 4, pp Chorn, A.J., 1973, Numercal Study of Slghtly Vscous Flows, Jounal of Flud Mechancs, v. 57, pp Greengard, C., 1985, The core spreadng vortex method aproxmates the wrong equaton, Journal of Computatonal Physcs, Vol. 61, pp Leonard, A., 198, methods for flow smulaton, Journal of Computatonal Physcs, Vol. 37, pp Lews, R.I., 1991, Element Methods for Flud Dynamc Analyss of Engneerng Systems, Cambrdge, Cambrdge Unversty Press. Ross, L., 1996, Ressurectng core spreadng vortex methods: A new scheme that s both determnstc and convergent, SIAM Journal, Vol. 17, No., pp Ross, L.F., 1997, Mergng computatonal elements n vortex smulatons, SIAM J. Sc. Comput., vol. 18, nº 4, pp Santago, V.S., Bodsten, G.C.R., Estudo Comparatvo de Quatro Dferentes Métodos de Dfusão Vscosa para Aplcação no Método de Vórtces, Paper CIT6-49, Anas do XI Congresso Braslero de Engenhara e Cêncas Térmcas - ENCIT (em CD-ROM), Curtba, PR, ISBN , 5 a 8 de dezembro, 6, pp Santago, V. S., Slva, D. F. C., Bodsten, G. C. R., Análse de Desempenho do Método de Expansão em Multpolos Adaptatvo Aplcado a Smulações Numércas Va Método de Vórtces, Paper CIT6-48, Anas do XI Congresso Braslero de Engenhara e Cêncas Térmcas - ENCIT (em CD-ROM), Curtba, PR, ISBN , 5 a 8 de dezembro, 6, pp RESPONSIBILITY NOTICE The authors are the only responsble for the prnted materal ncluded n ths paper.