FINITE VOLUME SIMULATION OF A FLOW OVER A NACA 0012 USING JAMESON, MacCORMACK, SHU AND TVD ESQUEMES

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1 FINITE VOLUME SIMULATION OF A FLOW OVER A NACA 00 USING JAMESON MacCORMACK SHU AND TVD ESQUEMES Oscar Aras a Oscar Falcnell b Nde Fco Jr. a and Sergo Elaskar bc a Insttuto Tecnológco de Aeronáutca Brasl e-mal: nde@ta.br b c Unversdad Naconal de Córdoba. CONICET Argentna Keywords: Fnte volume Jameson MacCormack Shu TVD NACA00. Abstract. In ths paper s presented the results obtaned by means numercal smulatons of the flow over the NACA 00 arfol at dfferent Mach numbers usng two codes: one developed at Insttuto Tecnológco de Aeronáutca of Brazl (ITA); the second developed at the Departamento de Aeronáutca de la Unversdad Naconal de Córdoba (UNC). The computer code developed at ITA works wth a boundary-ftted structured-grd context; and t solves the equatons by a fnte-volume technque usng three dfferent tme-marchng schemes. The Euler flow was modeled as well as a Reynolds-averaged Naver Stokes formulaton was calculated. Two Euler cases were done usng the Jameson MacCormack and the Shu schemes to advance n tme. The computer code developed at the UNC uses a Total Varaton Dmnshng (TVD) scheme mplemented n a non structured fnte volume formulaton for solvng the 3D Euler equatons s presented. Fnally the pressure coeffcent dstrbuton along the arfol chord and the contours of the flow propertes were compared wth other data avalable n lterature.

2 INTRODUCTION The numercal smulatons comparson of the flow around the NACA 00 arfol at dfferent Mach numbers usng two codes are presented n ths paper. The frst code was developed at Insttuto Tecnológco de Aeronáutca - ITA and the second software was developed at the Unversdad Naconal de Córdoba - UNC. Both computatonal codes mplement a fnte volume method to solve the Euler equatons. The ITA code works on two-dmensonal structured grd and t posses the capacty to work wth three dfferent schemes: () the Jameson scheme (Jameson et al 98) usng a fve stage Runge-Kutta tme ntegraton; () the MacCormack scheme (MacCormack 984) based upon the predctor and corrector strategy to advance n tme; () and fnally the Shu scheme (Yee 997) whch uses a varaton of the Jameson tme ntegraton n order to better capture of shock waves. The UNC code works on three-dmensonal unstructured grd and t mplements three dfferent numercal schemes: () the frst order flux vector-splttng () the Harten-Yee TVD scheme (Harten 983; Yee Warmng and Harten 985) () the mproved Harten-Yee TVD technques developed at the UNC (Falcnell et al 007). Furthermore the computatonal effort and total CPU tme related to the fnte-volume schemes mplemented s compared. The flow over the NACA 00 at transonc and supersonc veloctes was solvng to calculate the general characterstcs of each scheme. DESCRIPTION OF THE ITA CODE The descrpton of the utlzed schemes and the boundary condtons s presented n ths secton. Ths code was developed to solve aerodynamcs flows.. Jameson scheme The method developed by Antony Jameson and coworkers (Ortega 995; Pullam 986) ncorporates effcent stablty characterstcs and hgh compatblty to multgrd smulatons. It apples a fve-stage Runge-Kutta to advance the soluton n tme. The method s fourthorder accurate n tme and second-order n space. The fluxes across the grd-cell surface are calculated by averagng the flow propertes on both sdes gvng rse to a centraldscretzaton scheme. Therefore t s ntrnscally a central dfference approxmaton n space. As a consequence t requres the use of addtonal numercal dsspaton terms n order to guarantee stablty. In ths case a non-lnear artfcal vscosty s used (Jameson et al. 98). To advance the soluton n tme Jameson suggested a fve-stage Runge-Kutta ntegraton scheme. The numercal dsspatve terms are evaluated ust n the frst two stages (Mavrpls 988). The non-lnear numercal dsspaton has a pressure sensor that s somewhat more complcated than the one used here (Ortega 995). Therefore the author decded to calculate these dsspatve fluxes n all the Runge-Kutta stages to mprove the stablty. Thus addng the numercal dsspaton operator one gets: dq = 0 ( ) ( ) Te Q Da Q dt V () where ( ) Da Q denotes the artfcal vscosty. Let the superscrpt n denote the tme level. Thus n represents the next tme level after a tme ncrement equal to Δ t. To advance the calculaton towards the steady state soluton

3 one wrtes: 0 n Q Q = () e t Q Q T Q Da Q V α Δ = () () 0 e t Q Q T Q Da Q V α Δ = e t Q Q T Q Da Q V α Δ = e t Q Q T Q Da Q V α Δ = e t Q Q T Q Da Q V α Δ = 5 n Q Q =. where the subscrpts n the Q vector where neglected for smplcty. The standard values for the α coeffcents used n the present work are: 4 α = 6 α = α = 4 α = and 5 α =.. MacCormack scheme The explct MacCormack algorthm (MacCormack 984) uses two steps to advance n tme one s the predctor and the other s the corrector. It s a second-order-accurate n both space and tme. In the predctor step the flux vector at a certan face s calculated usng the propertes at the forward cell whereas n the corrector step the propertes values to be used are the ones relatve to the backward cell. The dscrete approxmaton of all the fluxes crossng the surface of the control volume for the predctor step s: = n p S P S P S P S P Q T and for the corrector t s:. = n n n n n c S P S P S P S P Q T () (3) (4)

4 Consderng the forward and backward dscretzaton the scheme s as follows. Predctor: Corrector: Update: () Δt Q Q T Q Da Q = p V () Δt () Q Q T Q Da Q = 0 c V ( n ) () ( ) Q = Q Q. (5) (6) (7) The explct MacCormack scheme even though t s an upwnd method does not mplctly ntroduce a suffcent amount of artfcal dsspaton. Thus t s necessary to explctly add t. Wthout these terms the code dverges..3 Shu scheme Another procedure to advance the soluton n tme utlzed n ths work s that proposed by Shu (Shu 989) and explaned by Yee (Yee 997). The resultant numerc method has good shock wave capturng characterstcs. It uses a varant of the Runge-Kutta ntegraton that needs only three steps: () Δt Q Q T Q Da Q = e V () 3 () Δt () Q Q Q T Q Da Q = 0 e 4 4 4V (8) ( n ) Δt Q Q Q T Q Da Q. = 0 e 3 3 3V.4 Boundary condtons The types of boundary condtons used for ths case are shown n Fgure. The wall far feld and perodc boundary condtons are descrbed next: o Wall Boundary condton: In the Euler case the non-penetrablty condton s used forcng the flow to be tangent to a sold surface see Fgure.

5 Fgure : Boundary Condtons for an O-grd. Fgure : Velocty Vectors next to a Sold Surface for Euler Formulaton. Thus accordng to Fgure the velocty componentsu 0 and V 0 n the x and y drecton respectvely for the ghost volume must be U = U n n V n n 0 y x x y V = V n n U n n 0 x y x y where the unt vector components n the x- and y-drecton n x and n y are (9) (0)

6 and n x ( S ) x / = d n y ( S ) y / = d () ( x) ( y) d = S S / / / () s the magntude of the surface vector. o Far feld boundary condton: The dea s to attrbute free-stream values to all the flow propertes at the external boundary. It s mportant to pont out that non-reflectve boundary condtons were not mplemented o Perodc boundary condton: ths could be consdered as a dfferent type of symmetry condton. The dea s to set the flux of all varables leavng the perodc outlet boundary equal to the flux enterng the perodc nlet boundary. Therefore for any property φ of the flow on a O-grd ths condton s establshed as φ φ φ φ = φ max = = φ max = = φ = 0 max = φ = max Fnally t s mportant to pont out that the ntal condtons necessary to start the teraton process were the free-stream values over the entre computatonal doman..5 Valdaton test cases for the ITA code To valdate the correct behavor of the ITA code are carred out numercal smulatons of the nvscd flows over the NACA 00 arfol. The results are compared wth other results avalable n the lterature. The numercal smulatons were carred out on a Dell computer Lattude D600 Pentum V wth a 700-Mhz processor and 5 MB of RAM. The numercal soluton convergence was checked calculatng the maxmum densty varaton over the entre 9 computatonal doman. If t was smaller than0 the program stops. For comparatve purposes the schemes of Jameson MacCormack and Shu have been used to solve transonc and supersonc flow over the NACA 00 arfol. For the transonc case the Mach number s M = 0.8 and α =.5. For the supersonc case the Mach number s. at zero angle of attack. All numercal smulatons were done wth a boundary-ftted structured grd usng 89 x 53 elements n the and drecton respectvely. The pressure coeffcent along the arfol and the convergence rate were evaluated. The schemes were run wth the maxmum CFL number possble. Thus the CFL number was equal to.7 for the Jameson method 0.4 for MacCormack and 0.6 for the Shu formulaton. The non-lnear artfcal vscosty coeffcents for both cases were K = and K 4 =

7 .6 Mach 0.8 Alpha.5 All The pressure coeffcents of the dfferent schemes are shown n Fgure 3. All the results presented by ths code looks smlar as that provded by Pullam (Pullam 986) usng a fnte dfference method and Jameson-Mavrpls usng a regular trangular mesh. Fgure 3: Pressure Coeffcent Comparson for Jameson McCormack and Shu Schemes. The sucton peak at the upper surface was found to be approxmated -. for our present smulatons comparng wth the Pullam soluton. However for all the results ncludng that one of Pullan the upper shock wave was shfted towards the leadng edge when compared wth the Jameson-Mavrpls soluton as reported by Wenneker (Wenneker 00) on hs doctoral thess. Ths could happens because of the entropy generaton at the leadng edge whch creates a numercal boundary layer causng losses (close to the arfol) that are vsble n the constant Mach lnes see Fgure 4. The weak shock wave at the lower part of the arfol s partcularly hard to capture but all the schemes dd t good. However there was a dfference n the lower shock when compared wth Pullam. Jameson predcts the shock wake formaton before Pullan whle Shu s capture the shock almost at the same poston of Pullam. MacCormack predcts a stronger aft postoned shock. Ths dfference may be related to two possble reasons: one s the AV coeffcents combned wth the type of grd used by Pullam and the other s the use by Pullan of upwnd dfferencng n supersonc regons as suggested by Steger (Steger 978) producng better shock capturng capabltes. However t should be ponted out that the Cp mnmum value around was the same for all the schemes.

8 Examnng the convergence hstory n Fgure 5 the dfferences start to show up. Durng the frst 000 teratons the Jameson formulaton showed a hghly oscllatory behavor but after 400 these oscllatons were perodc untl the convergence crteron was acheved. The MacCormack mplementaton had an ntal behavor smlar to that of Jameson. However t took more than three tmes more teratons (60) to attan the convergence crteron. For Shu scheme the frst 000 teratons showed a sgnfcant resdue drop. Unfortunately as the computaton contnued the densty varaton rased agan. Just over 7000 tme steps the resdue looked stable n ts way to convergence but the very tght convergence crteron was reached after 3506 teratons a number hgher than the other schemes. Y X Fgure 4: Mach contours usng the Jameson scheme For a better understandng of these results the computatonal effort ξ was calculated. Accordng to Beam and Warmng (Beam and Warmng 978). ξ = I J CP Number of tme steps (3) where I and J are the number of grd ponts along each drecton CP s the computatonal tme n seconds and the number of steps s the total number of teratons. The results were putted together n Table I: The MacCormack method generated 6.70% less computatonal effort than the Jameson method. Ths was because a predctor-corrector procedure was used to advance the soluton n tme. Thus there were only two steps per teraton. The Shu scheme demanded 4.3% less effort than Jameson because t used a three-step Runge-Kutta. Ths s a varaton from the orgnal fve step Runge-Kutta formulaton. In regard to the CPU tme process a more realstc fgure of mert Jameson was 4.9% faster than MacCormack and Shu took almost three tmes longer than Jameson.

9 Fgure 5: Comparson of densty varaton hstores. CFLmax Comp. Tme Cp (sec) Iteratons Comp. Effort ξ (sec) Jameson x0 MacCormack x Shu x Table I: Computatonal effort for dfferent schemes. The man dea of usng the Shu varant was to mprove the results partcularly n respect to a better shock wave resoluton. Although the general pressure dstrbuton wth ths technque s almost the same as the others there s an mprovement n the lower shock wave poston. Ths scheme shows the closets results to the dstrbuton presented by Pullam. No other mprovement was obtaned by the present author. The MacCormack scheme although t took 9.8% more tme than the Jameson scheme showed a consderable lower computatonal effort. Ths suggests that t maybe adequate for very fne mesh lke the ones used for vscous smulatons.

10 .7 Mach. Alpha 0 A supersonc flow M =. over the NACA 00 at zero angle of attack was smulated usng the same 89 x 53 grd of the transonc calculaton. For ths case the CFL values were the same as the prevous case for all the schemes. In fgure 6 the results are compared wth those presented by Yoshhara & Sacher (Yoshhara and Sacher 985) and Wenneker (Wenneker 00). The crcles correspond to the Yoshhara soluton. The delta symbols ndcate the results presented by Wenneker usng an unstructured grd wth 344 cells. The Cp dstrbuton found usng the three dfferent schemes s n very good agreement wth the lterature except for a very mld asymmetry almost mperceptble located close to the leadng edge. Fgure 6: Pressure Coeffcent Comparson for M =. and α = 0.0. Comparson of the constant Mach lnes of the Jameson scheme wth the Wenneker result s shown n Fgure 7. The result obtaned by the present code was smlar to that presented by that researcher. However the bow shock dd appear to be thcker. Further Yoshhara and Sacher found the shock poston to be at x/c = from the leadng edge.

11 (a) (b) Fgure 7: Mach Isolnes Comparson from (a) Wenneker and (b) ITA code. (c) Mach Contours. The shock poston data s condensng n table II. Wenneker obtans whle the Jameson calculaton gets a value almost exactly to that of Yoshhara. The MacCormack and Shu formulatons provded both a hgher error of an order of 5 %. Ths could be produce not only due to the dfferences n the schemes t could also be due to the fact that reflectve boundary condtons are not mplemented here and as the smulaton advance n tme there s more possblty to contamnated the arfol wth returnng dsturbances form the far feld boundary. Ths could be possble especally for the schemes that take more tme to converge. (c)

12 Shock Poston Yoshhara & Sacher Wenneker Jameson MacCormack Shu Error (%) Table II: Shock wave poston. Fgure 8: Comparson of densty varaton hstores. CFLmax Comp. Tme Cp (sec) Iteratons Comp. Effort ξ (sec) Jameson x0 MacCormack x Shu x Table III: Computatonal effort for dfferent schemes. The densty resdue s shown n Fgure 8 and the computatonal effort s shown n table III. It s clear that agan the MacCormack scheme has the lower computatonal effort 56% lower than Jameson and Shu was close to Jameson showng 7% less effort

13 3 DESCRIPTION OF THE UNC CODE The three-dmensonal Euler equatons can be wrtten as: U F = 0 t where U s the vector of conservatve varables and F s the 3D vectoral flow. The temporal change of the conservatve varables can be wrtten as: (4) nfaces n n Δt * Vol = U = U F n A = 0 (5) The numercal flux proecton n the drecton normal to the face F n= * * f / s calculated usng the TVD scheme proposed for Harten (Yee 997; Harten 983) and modfed by Yee (Yee Warmng and Harten 985). Although ths method was orgnally developed for fnte dfferences technques t has been extended successfully for fnte volume formulatons smlar to the one used n ths work (Udrea 999). The Eq. 5 mples the use of a locally algned system of coordnates whose unt vector concdes wth the normal to the face of the cell and the unt vectors and k are tangental drectons. Snce the local Remann problem s solved wth rotated data the egensystem s calculated n the locally algned coordnate frame. The numercal flux can then be expressed as: * f f m m / = Φ / r / m f (6) where f and f are the physcal fluxes normal to the face n each cells m egenvector and Φ / s n the orgnal Harten-Yee scheme defned as: m r / s the m-th rght beng: m m m m m m / g g / / / Φ = λ γ α (7) g ( / / S / /) S = max 0 mn λ α λ α m m m m m (8) ( m / ) S = sgn λ ; m m g g m s α / 0 m / m α γ / = m 0 s α / = 0 (9) m where α / s the ump of the conserved varables across the nterfaces between cells and m and λ / s the m-th egenvalue of the Jacoban matrx.

14 The lmter functon gven n Eq. (8) s known as mnmod (Sweeby 984; Leveque 99). The mnmod selects the mnmum possble value so that the scheme s TVD. The other end s the lmter functon superbee (Hrsch 99) that ponders the contrbuton of the hgh order flux. The mplementaton of ths functon leads to an excessvely compressve scheme whch t s not very robust for general practcal applcatons. In the numercal soluton of the three-dmensonal Euler equatons fve wave famles appear. If the fve wave famles are enumerated n correspondence wth ther speed beng one the slowest and fve the faster t can be demonstrated that for waves of the famles two to four the characterstc veloctes at both sdes of the dscontnuty are the same and equal to the velocty dscontnuty (Hrsch 99; Toro 999). Ths property makes very dffcult to solve theses waves accurately unless they are solved dffusely. In ths work t s explored the possblty of mplementng dfferent lmter functons for dfferent wave famles. The obectve s to mprove the numercal resoluton of the dscontnutes assocated wth the famles two to four usng compressve lmter functons (superbee) and wthout losng robustness due manly to the use of dffusve lmter functons (mnmod) for the wave famles one and fve. To ntroduce n the numercal fluxes calculatons the lmter functon superbee the Eq. (8) s replaced by the followng expresson: g m 0 f α α < 0 = m m / / m m m m max 0mn ( r )mn ( r ) λ / α / f α / α / 0 (0) m m m m beng: r / / / / / =λ α λ α. To mprove the overall scheme robustness the mplementaton of dfferent lmter functons s carred out only n those cells nterfaces where the greater relatve ntenstes of the dscontnutes n central waves are regstered and usng the conventonal Harten-Yee TVD scheme n all other cases (Falcnell Elaskar and Tamagno 007). 3. Boundary condtons The treatment of the boundary condtons s carred out through the magnary cells technque (Hrsch 99; Toro 999). Fve dfferent types of boundares are consdered: Subsonc nlet - Supersonc nlet 3 Subsonc ext 4 - Supersonc ext 5 - Non penetraton (sold boundary and symmetry).

15 3. Valdaton test cases for the UNC code In order to verfy the accuracy and robustness of the proposed scheme two test cases are smulated. The frst one s the flow nsde of a shock tube. Ths example s used to explore the capacty of the scheme whch has been descrbed to model contact dscontnutes; the flow does not have velocty regardng the contact dscontnuty (dscontnuty n waves of the famly ). The second test s the smulaton of a slp layer between two flows wth dfferent veloctes and denstes and equal pressures. Ths test was chosen to study the capacty of the scheme to solve flows n whch the dscontnuty s n the velocty tangent to a gven nterface (dscontnuty n waves of the famles 3 and 4). The numercal smulatons of the UNC code were realzed on notebook PENTIUM IV de.60ghz con 48MB de RAM 3.3 Flow nsde a Sock tube The shock tube has a rectangular secton of 4cm x 4cm and a length of m; the gas s ar. The drver secton s flled wth atmospherc ntal condtons. The pressure n the drven secton s 0. atmospheres and possesses the same temperature than the drver. The analytcal soluton s gven by a three waves system a shock wave that advances toward the rght at 543.4m/s a contact dscontnuty that also moves toward the rght at 77.6m/s and an expanson fan whose head wave travels to the left at 4.9 m/s and the tal wave travelng at 338. m/s leftward also 8. The computatonal mesh s obtaned by means of a generatng cell bult up by 6 tetrahedrons that form a cube of cm sde length. The cube s repeated up to complete an arrangement of 4x4x00. Fgure 9 shows the densty as a functon of (x/t). In ths fgure are compared: the analytcal soluton the results obtaned applyng the new TVD scheme and those obtaned wth the conventonal Harten-Yee TVD scheme. It can be apprecated that capture of the contact dscontnuty has been mproved notably. Furthermore the absence of oscllatons next to dscontnutes can clearly be seen as well as the accuracy acheved wth the speed of the waves. 3.4 Slp surface The second test case s an ar layer wth a densty of.5kg/m3 a velocty of 690.3m/s and a pressure of 00000Pa that flows over another layer wth a densty of 3Kg/m3 a velocty of 648.m/s and a pressure of 00000Pa. The analytcal soluton predcts the slp of a flow on another wthout nterferences. However due to the numercal vscosty the computed soluton produces an apparent mxture zone that gets wder downstream the end of the spltter plane. The spreadng of such unphyscal mxng regon quantfes the lack of accuracy of the numercal method. In ths second case the mesh has tetrahedrons and.503 nodes. The control volume where the flow develops s 5m long m hgh and m wde. The non-structured mesh does not have any bas plane that may nfluence the locaton of the slp dscontnuty.

16 .4. superbee lmter functon n waves 3 and 4 and mnmod lmter functon n waves and 5 mnmod lmter functon n all waves exact soluton Densty [Kg/m^3] The boundary condtons at the nlet are: x/t [m] Fgure 9. Shock tube results. For x = -m.5m; -0.5m y 0.5m; 0m z 0.5m the condtons are: ρ = 3 Kg/m 3 u = 648.m/s. For x = -.5; -0.5m y 0.5m ; 0.5m z m are mplemented: ρ =.5 Kg/m 3 u = 690.3m/s. For both case t s verfed that v = w = 0 p = Pa In Fgure 0 s presented the absolute vale of the % error dstrbuton n the velocty predcted at a lateral plane of the computatonal doman. The results usng the Harten-Yee TVD scheme are shown n the upper half and the results obtaned wth the new scheme are shown n the lower half. The error s e% 00 n / = u ut z u. Beng u n the velocty calculated numercally; u t( z) = 690.3m/s the velocty f the z-coordnate of the center of gravty (CG) of the cell s hgher than 0.5m or u t( z) = 648.m/s f the cell CG z-coordnate s smaller than 0.5m; u = m/s 648.m/s = 69. m/s. The obtaned numercal solutons do not show apprecable varatons as functon of the computatonal doman wdth; the solutons for any plane are representatve of what happens n the whole doman. For ths reason the results shown n Fgure 0 are only those taken n the plane y = -0.5m. As t can be seen the angle formed by the straght lnes that lmt the area wth errors hgher than % s 9.9 o for the Harten-Yee scheme whle that angle reduced to 4.8 o for the new scheme. Consequently the proposed scheme s notably less dffusve.

17 Fgure 0. Slp surface results. Error: whte 0% e % green % e 5% yellow 5% e 0% lght blue 0% e 0% red 0% e. 4 CODES COMPARISON In ths secton are presented the numercal results obtaned by means of the ITA and UNC codes. The smulatons carred out by the ITA code consder two dmensonal flow and structured meshes. The numercal smulatons obtaned wth the UNC code consder three dmensonal flow and non-structured meshes; the used fnte volumes are tetrahedrons wth four nodes. Four tests were realzed: NACA 00 M = 3 α = 0 o NACA 00 M = 7 α =.5 o 3 NACA 00 M = 8 α =.5 o 4 NACA 00 M =. α = 0 o To make comparable the results the 3D meshes used for the UNC code and the D meshes mplemented at the ITA code posse equal number the nodes over the arfol. For the UNC code the numercal results were obtaned usng the conventonal Harten- Yee TVD scheme. Fgures and 3 show that the ITA codes posses a better behavour prncpally n transonc regme (M = 0.8). For ths case the UNC predcts the shock wave formaton wth a 0% error. However the UNC code reproduces wth more accuracy the sharper form of the shock wave. For the supersonc flow (M =.) both codes reproduce correctly the Cp dstrbuton (see Fgures 6 and 4). The results obtaned by UNC codes correspond to 3D smulatons. It can be apprecated from Fgures 3 and 4 that the scatterng of the results at each x-staton s very small compared wth related local dynamc pressures.

18 -0.6 Cp x/c UNC code ITA code 0.8 Fgure. Comparson for arfol NACA 00 M = 0.3 α = 0 o Cp x/c UNC code ITA code Fgure. Comparson for arfol NACA 00 M = 0.7 α =.5 o 5. CONCLUSIONS The man task was to smulate usng two codes the nvsd flow over the NACA 00 arfol consderng dfferent Mach numbers (M = 0.3; 0.7; 0.8;.). The frst code developed at ITA and the second at the UNC. Both codes use fnte volume to solve the Euler equatons however have dfferent obectves structures and technques. The ITA code was developed to smulate subsonc and transonc aerodynamcs flows; the UNC code posses the obectve to smulate supersonc and hypersonc flows. The codes has been valdate n functon of ther obectves.

19 The man concluson s that the ITA code presents better behavor to predct the pressure dstrbuton over the arfol for subsonc and transonc flows. For supersonc flows (M =.) the numercal results of the two codes were very good. However the UNC captures more accurately the sharper form of the shock wave (M = 0.8). Theses conclusons are n concordance wth the obectves for the both codes. The followng step s to carry out comparson for hgh supersonc flows. Cp x/c UNC ITA.5 Fgure 3. Comparson for arfol NACA 00 M = 0.8 α =.5 o Fgure 4. Comparson for M =. α = 0 o. Blue: Wenneker. Green: AGARD. Red: UNC code. Fnally the smulatons wth the ITA code revealed that the Jameson scheme although demandng more computatonal effort than the MacCormack s generated much better results at least for the NACA 00 cases studed. Moreover Jameson s scheme allows for greater CFL numbers whch could at least n part compensate the larger number of teratons t needs to converge as well as the more computer ntense calculatons between tme steps. Furthermore t was not possble to run the MacCormack scheme wthout artfcal vscosty terms despte ts upwnd character.

20 REFERENCES Beam Rchard; Warmng R. F. An mplct factored scheme for the compressble Naver- Stokes Equatons. AIAA Journal 6(4): Aprl 978. Falcnell Oscar Elaskar Sergo and Tamagno José Reducng the Numercal Vscosty n Non Structured Three-Dmensonal Fnte Volumes Computatons. AIAA Journal of Spacecraft and Rockets submtted 007. Harten A. Hgh resoluton schemes for hyperbolc conservaton laws Journal of Computatonal Physcs 49: Hrsch C. Numercal Computaton of Internal and External Flows Vol. Computatonal Methods for Invscd and Vscous Flows John Wley & Sons Ltd. London 99. Jameson A.; Schmdt W.; Turkel E. Numercal soluton of the Euler Equatons by fnte volume methods usng runge-kutta tme-steppng schemes. In Flud And Plasma Dynamcs Conference Palo Alto. Palo Alto: AIAA 98. Paper No.8-59 Leveque R. J. Numercal Methods for Conservaton Law Brkhäuser Verlag Basel 99. MacCormack R. W. An ntroducton and revew of the bascs of computatonal flud dynamcs. Washngton DC: Unversty of Washngton 984. (Lectures notes) Mavrpls D. J. Multgrd soluton of the two-dmensonal Euler equatons on unstructured trangular Meshes. AIAA Journal 6(7): Ortega M. A. Apostla do curso de CFD. São José dos Campos: Insttuto Tecnológco de Aeronáutca 995. Pullan Thomas. Artfcal dsspaton models for Euler equatons. AIAA Journal 4(): December 986. Shu Ch-Wang. Effcent Implementaton of Essentally Non-oscllatory Shock Capturng Schemes II. Journal of Computatonal Physcs 83: Steger J. L. Implct fnte dfference smulaton of flow about arbtrary geometres wth applcatons to arfols. AIAA Journal 6:679 July 978. Sweeby P.K. Hgh Resoluton Schemes Usng Flux Lmters for Hyperbolc Conservaton Laws SIAM Journal on Numercal Analyss : Toro E. F. Remann Solvers and Numercal Methods for Flud Dynamcs Sprnger-Verlag Berln 999. Udrea B. An advanced mplct solver for MHD PhD Thess Unversty of Washngton 999. Wenneker Ivo. Computaton of flows usng unstructured staggered grds. 00. Thess. (PhD Dssertaton) - DELFT Unversty of Technology Delft The Netherlands. Yee H. C. Explct and mplct multdmensonal compact hgh-resoluton shock capturng methods formulaton. Journal of Computatonal Physcs 3: Yee H.C. Warmng R.F. and Harten A. Implct Total Varaton Dmnshng (TVD) Schemes for Steady-State Calculatons Journal of Computatonal Physcs 57: Yoshhara H. Sacher P. Test cases for nvscd flow feld methods. AGARDograph No. AGARD Neully-sur-Sene France 985.