ICAS 000 CONGRESS PHANTOM VORTICITY IN EULER SOLUTIONS ON HIGHLY STRETCHED GRIDS S. A. Prnce, D. K. Ludlow, N. Qn Cranfeld College of Aeronautcs, Bedfordshre, UK Currently DERA Bedford, UK Keywords: Phantom vortcty, Numercal dsspaton, Euler equatons, Grd stretchng Abstract The excessve numercal dsspaton, whch s known to occur when the steady Euler equatons are solved on fne, hghly stretched grds s nvestgated. A new formulaton of the hgher order MUSCL scheme that accounts for grd non-unformty s derved and appled to the allevaton of ths phantom vortcty. Introducton Several nvestgators have shown that numercal solutons of the Euler equatons on hghly stretched grds past a smooth nclned body can exhbt a numercal boundary layer leadng to the development of symmetrc leeward sde vortces [][][3]. The development of ths phantom vortcty can occur even wth the applcaton of a slp boundary condton at the wall. The nvscd character of the Euler equatons means that they cannot account for vscous vortcty generaton. Numercal soluton of these equatons can, however, nvolve mechansms for the erroneous generaton of vortcty. In nvscd flow, curved shock waves ntroduce crculaton, entropy layers and vortcty nto the soluton. Marcon [] demonstrated that for a concal-euler soluton on a slender cone at angle of attack n a supersonc stream, a curved embedded crossflow shock causes separaton of the flow from the cone surface and a vortex to form near the leeward symmetry plane. In addton, snce Crown copyrght 000. Publshed by the Internatonal Councl of the Aeronautcal Scences wth the permsson of the Defence Evaluaton and Research Agency on behalf of the Controller of HMSO numercal solutons are obtaned from approxmate or dscretsed forms of the governng equatons, Euler solutons may nclude vortcty generaton by the effect of truncaton error numercal dsspaton. Once a vortex s formed the Euler equatons allow for ts convecton downstream but cannot smulate ts dffuson due to physcal vscosty. All computatonal schemes, however, are dsspatve and even wthout the addton of artfcal vscosty, wll dffuse and destroy vortcty. Numercal dsspaton wll dmnsh the strength of vortces, but not n a manner representatve of a vscous flowfeld. The accurate capture of boundary layer separaton and subsequent vortex development s crtcal for the computatonal predcton of the flow past smooth arcraft or mssle forebodes. If the errors assocated wth phantom vortcty are large enough, wll they lead to errors n vscous calculatons, and f so, what can be done to allevate ts effect? Ths paper seeks to address these ssues by detaled numercal studes of the supersonc flow about an nclned slender body, and by careful nvestgaton of the soluton algorthms used n the flow solver. Prevous Investgatons Chnlov [4] nvestgated the phenomenon of the non-physcal, numercal boundary layer that he found to develop near the body surface when a supersonc nvscd stream flows past a blunt body. A steady two-dmensonal supersonc flow past a crcular cylnder was nvestgated usng several mesh resolutons by use of a frstorder fnte volume numercal Godunov scheme. Chnlov found that he could reduce the 43.
S. A. Prnce, D. K. Ludlow, N. Qn phantom vscosty by ether refnng the grd or employng second-order spatal dscretsaton. In order to compute a flow soluton effcently and to resolve the boundary layer and assocated flow features, grd stretchng s often used. Typcally a grd s created whch s stretched n a wall normal drecton such that the cell wdth s a strctly ncreasng functon of the dstance from the wall. Although the prevous nvestgaton was performed wthout any apprecable grd stretchng, a number of nvestgators have shown that grd stretchng can have a bg effect on the accuracy of modern schemes for compressble flud flows. Turkel [5] showed that many popular central dfference and upwnd methods reduce to frst-order spatal accuracy n regons where the grd s hghly non-unform. Varous algebrac and exponental grd-stretchng functons were nvestgated. It was found that algebrac stretchng was suffcently smooth to allow second-order technques to mantan ther formal accuracy. Exponental stretchng functons however proved to deterorate the spatal accuracy to frst-order unless specal weghted formulas are used. In addton t was demonstrated that second-order accuracy could be mantaned wth these schemes only f the mesh accurately reflects the propertes of the soluton. A number of methods have been developed to account for grd stretchng and thereby mantan spatal accuracy. Batna [6] developed mproved algorthms for spatal and temporal dscretsaton n hs unstructured Euler solver for the nvestgaton of the unsteady aerodynamcs of a two-dmensonal ptchng aerofol. In partcular he developed a smple nterpolaton of the prmtve varables that he employed n the standard MUSCL scheme n order to treat hghly skewed tetrahedral cells. Ths weghted nterpolaton of the prmtve varables was based on the dstance between the centrod and the mdpont of the approprate edge. Batna [7] further developed ths technque for three-dmensonal flows, ths tme wth a dfferent MUSCL type scheme. Lou and Hsu [8] developed a hgh resoluton scheme for ther tme accurate threedmensonal structured fnte volume solver based on Roe s upwnd technque for flux dfference splttng. Non-unformty of cell szes was accounted for by dervng a number of factors based on the szes of each cell n the stencl. Wth ths nformaton t was decded to nvestgate the generaton of phantom vortcty around a slender, sharp nosed, smooth cylndrcal body. In partcular, the affect of grd stretchng and the spatal accuracy of the solver were studed. 3 Numercal Solver and Grd The numercal analyss was performed usng a three-dmensonal tme-marchng, cell-centred fnte volume Naver-Stokes (NS) solver that was operated n Euler mode. The steady compressble Euler equatons were solved usng a cell-centred fnte volume approach wthn a structured dscretsed doman. Invscd fluxes were calculated usng Osher s approxmate Remann solver. Hgher (second or thrd) order spatal accuracy was attaned by the use of the MUSCL varable extrapolaton together wth a slope lmter. An approxmate soluton of the Remann boundary problem was used to prescrbe the Euler slp boundary condtons at the wall. More detals about the solver can be found n [9]. A three-calbre tangent-ogve cylnder geometry was chosen for the current numercal nvestgaton. Expermental studes wth ths geometry nclned at 0 degrees to a Mach.0 flow has been carred out at ONERA [0]. The Expermental results revealed a well-developed symmetrc vortex pattern on the leeward sde of the body. Fve sngle-block structured grds of varyng cell number, were generated around the body. The coarsest Euler grd was of sze 60 54 45 and employed near-wall cells of 0.0085D radal thckness (D= calbre) and a tanh radal stretchng functon. The next four grds, sutable for NS calculatons, all used near wall cells of 0-4 D radal thckness and agan employed a tanh radal stretchng functon. The 43.
PHANTOM VORTICITY IN EULER SOLUTIONS ON HIGHLY STRETCHED GRIDS cells n each grd were spaced unformly n the crcumferental drecton. 4 The Effect of Spatal Accuracy and Grd Stretchng. The NS solver was employed n Euler mode to obtan solutons on the fve grds (detaled n the key n Fgure ) usng the standard MUSCL formulaton wth a κ factor set for thrd-order spatal accuracy. It was found that the frst two grds converged well down to fve orders of magntude, whereas those on the three fnest grds stalled at around to.5 orders of resdual convergence. The CFL number, ntally set to 0.3, was successvely reduced to 0.05 n an attempt to converge the solutons further, but was found to have lttle effect. Analyss of the stalled solutons, checkng every 000 tme steps, revealed that the flow structure dd not exhbt any apprecable change. Fgure presents the fve solutons for the crcumferental pressure dstrbutons at x/d=7. It would be expected that, as the grd s further refned, the soluton would converge to the same crcumferental dstrbuton. What s observed, however, s a progressve devaton from the expected soluton (curve for grd ) and the development of pronounced sucton peaks due to the resoluton of phantom vortces on the leeward sde of the body. p C 0.0 0.0-0.0-0.04-0.06-0.08-0. -0. Euler Grd (60 x 54 x 45) NSGRID (33 x 33 x 33) NSGRID (60 x 70 x 45) NSGRID3 (60 x 70 x 73) NSGRID4 (60 x 85 x 73) -0.4 0 0 40 60 80 00 0 40 60 80 Ph (deg) Fgure : Grd convergence for Euler solutons on fve grds usng thrd-order MUSCL scheme. Fgure presents the computatonal grd and densty contours at x/d=8 obtaned by the fully converged Euler soluton on grd. Ths s the correct flow structure expected of an Euler soluton, wthout any flow separaton or leeward sde vortces. Close nspecton of the crossflow velocty vectors, however, dd reveal the presence of a slght velocty profle close to the leeward body surface. Fgure : Densty contours for thrd-order MUSCL Euler soluton on grd. The correspondng flow soluton on the fnest NS grd (60 85 73) at the same axal staton s presented n Fgure 3. Fgure 3b) clearly shows a well-developed numercal boundary layer that separates off the leeward sde of the body to form a prmary, secondary and tertary vortex system smlar to that expected of a vscous soluton rather than an Euler soluton. Snce the orgn of the vortcty s nonphyscal the resultng vortex pattern does not agree wth ether experment or wth the lamnar NS soluton on the same grd. Solutons were also obtaned usng a dfferent formulaton of the Euler wall slp boundary condton and usng the Roe Approxmate Remann Solver. None of these changes had any apprecable affect on the soluton. The next step n the nvestgaton was to look at the effect of spatal accuracy. By alterng the κ factor n the MUSCL scheme one can obtan second-order spatal accuracy, and by 43.3
S. A. Prnce, D. K. Ludlow, N. Qn swtchng off the MUSCL scheme one can obtan solutons of frst-order spatal accuracy. Fgure 4 presents the frst-order soluton at x/d=8. A small numercal boundary layer s stll observed on the leeward surface and the flow s seen to separate formng a very small, hardly vsble, vortex close to the leeward symmetry plane. Fgure 5 compares the crcumferental surface pressure dstrbuton at x/d=7 for the frst and thrd-order accurate solutons on grd 5. The hgher order result predcts separaton at around φ=00 o and two sucton peaks assocated wth a prmary and secondary vortex. The frstorder soluton s much closer to that expected of an Euler soluton, wth only a slght nflexon at φ=50 o ndcatng a small weak prmary vortex. a) Densty contours and crossflow plane grd a) Densty contours and crossflow plane grd b) Crossflow velocty vectors Fgure 3: Euler thrd-order MUSCL soluton on grd 5. b) Crossflow velocty vectors Fgure 4: Euler frst-order MUSCL soluton on grd 5. 43.4
PHANTOM VORTICITY IN EULER SOLUTIONS ON HIGHLY STRETCHED GRIDS The thrd-order value of the pressure coeffcent (C p ) at the leeward symmetry plane s equvalent to that predcted n a thrd-order vscous soluton, but that for the frst-order soluton s over-predcted by C p 0.0. Ths can be explaned by the overall reducton n spatal accuracy and the correspondng ncrease n numercal dsspaton. Fgure 5: Effect of spatal accuracy on Euler soluton on grd 5 A sxth grd was then generated n order to nvestgate further the effect of grd stretchng. Grd 6 (60 40 73), shown n Fgures 7 and 8 had a regon of cells of dentcal cell wdth adjacent to the wall; outsde these cells the cell wdth was ncreased smoothly, usng a tanh functon, to the freestream boundary. The nterface between the unform and stretched regons of the grd was postoned well beyond the nfluence of any boundary layer A thrd-order MUSCL calculaton was performed on grd 6. Whle the soluton no longer contaned a numercal boundary layer at the wall, a spurous numercal shear layer was seen to develop from the nterface where the cell wdth began to ncrease away from the wall. As wth the other computatons where phantom vortcty was found to develop, the soluton could not be fully converged. Grd 6 was then further modfed by pullng the grd stretchng nterface slghtly further away from the wall, ntroducng more cells nto the unform cell regon such that the cell wdth could be mantaned. The numercal shear layer was found to move wth the stretchng nterface. In effect, the numercal boundary layer was moved away from the wall and followed the poston of the dscontnuty n the gradent of cell wdth. Agan t was noted that the soluton could only be converged down to.5 orders before the calculaton stalled. Ths evdence suggests that the excessve numercal dsspaton s assocated wth the localzed loss of spatal accuracy when usng the standard hgh resoluton MUSCL scheme n regons where the grd s hghly stretched. In order to rectfy ths problem, t was decded to modfy the MUSCL scheme to account for grd stretchng. 5 A New Formulaton of the MUSCL Scheme for Non-Unform Grds Ths secton presents the dervaton of a hgh resoluton MUSCL scheme that uses a weghted formulaton of each cell sze n the computatonal stencl. The orgnal formulaton of the MUSCL scheme [] was derved from the pecewse quadratc dstrbuton for the varable U n a cell gven by: U = U 3κ + ( x x ) ( x x ) U x x U x where U s the average value defned by: + + U = U ( x) dx. x () The orgnal MUSCL scheme assumes that each cell s the same sze,.e., that there s no grd stretchng. Fgure 6 shows the one-dmensonal computatonal stencl about cell for a stretched grd. For a non-unform grd the term x n () 43.5
S. A. Prnce, D. K. Ludlow, N. Qn s equal to the wdth of cell denoted by s such that: s = x. () x + L + = U φ ( ) ( ) ( ) ) ~ + + ~ r + κφ + κφ + r r + r U + (6) The correspondng expresson for the values on the rght hand sde of the nterface + / (by a smlar treatment for the cell +) gves: Fgure 6: Fnte volume representaton of stretched grd about cell. Takng the functon U(x) to be a quadratc functon of x of the form: U U = A ( x x ) + B( x x ). (3) Substtutng x=x ± and U=U ± nto equaton (3) yelds two smultaneous lnear equatons whch can be solved for A and B. The dervatves of U wth respect to x at x=x are then evaluated by dfferentaton of equaton (3). Substtutng these dervatve terms nto equaton () and settng: r s s ± + ± = s ± = ± ± ± r ( U U ) (4) ~ (5) yelds an equaton, of the MUSCL form, for the evaluaton of the varables at the nterfaces of cell. For the left-hand sde of the nterface + /, ths results n the followng relaton: φ ( ) ( ) ( ) ) ~ + + ~ r + + κφ + + r + + κφ + r + r + + U R + = U + (7) where φ represents a flux lmter, whch for our study was that developed by Anderson et al []. Equatons (6) and (7) thus represent a modfed MUSCL scheme for non-unform grds, based on a quadratc dstrbuton of U across the cell. If the grd s unform then r ± =, and the orgnal MUSCL scheme s recovered. The modfed MUSCL scheme was employed on grd 6. Fgure 8 presents the crossflow soluton at x/d=8 and clearly demonstrates the dramatc mprovement the modfed MUSCL scheme produced. The numercal shear layer and the assocated vortces, whch dd appear as ntermedate solutons, were convected out of the soluton as t converged down by fve orders. The fnal soluton was comparable, even better, than the standard thrd-order MUSCL soluton on the Euler grd (grd ) despte the large cell stretchng. A further computaton was performed, employng the modfed MUSCL scheme on grd 5. Ths tme the numercal boundary layer and resultant phantom vortces appeared only as an ntermedate soluton and dsappeared as the soluton converged down to fve orders. A fnal test was carred out usng grd 5 to obtan a lamnar soluton wth both the standard and the modfed MUSCL scheme. The results, converged by fve orders, were practcally equvalent. Ths ndcates that the phantom vortcty s much less of a problem for vscous 43.6
PHANTOM VORTICITY IN EULER SOLUTIONS ON HIGHLY STRETCHED GRIDS calculatons; the physcal dsspaton beng much greater than numercal. the spatal accuracy of the scheme, hghly stretched grds, and to convergence problems. a) Ptot pressure contours and crossflow plane grd. a) Ptot pressure contours and crossflow plane grd. b ) Crossflow velocty vectors Fgure 7: Euler thrd-order soluton usng standard MUSCL scheme. 6 Phantom Vortcty: Conclusons The phenomenon of phantom vortcty n Euler solutons of the supersonc flow past smooth nclned slender bodes s lnked wth b ) Crossflow velocty vectors Fgure 8: Euler thrd-order soluton usng modfed MUSCL scheme. The phenomenon was found to be ndependent of the mplementaton of the Euler slp boundary condton and of the use of ether the Roe or Osher approxmate Remann solver. 43.7
S. A. Prnce, D. K. Ludlow, N. Qn Ths suggests that the phantom vortcty s generated by the excessve numercal dsspaton assocated wth the localzed loss of spatal accuracy n regons where the grd s hghly stretched. Schemes that do not account for non-unformty of the grd cannot mantan hgh order spatal accuracy and ntroduce excessve numercal dsspaton that s localzed to the hghly stretched regon of the grd. The Euler equatons, havng no mechansm for dffuson, cannot delocalze these errors whch can only convect through the flow-feld n the streamwse drecton. Snce the source of the numercal dsspaton hghly stretched grds wth an nconsstent scheme remans as the flow develop, more and more numercal dsspaton s generated and eventually the soluton cannot converge any further. Applcaton of a scheme that accounts for grd non-unformty wll mantan hgher order spatal accuracy n hghly non-unform regons of the mesh. The fact that phantom vortcty does occur but des away as the soluton s converged beyond three orders ndcates that there s stll a localzed loss of spatal accuracy, but the resultng numercal dsspaton s not strong enough to stall convergence. The equvalence of the lamnar solutons on a hghly stretched grd usng both formulatons of the MUSCL scheme shows that any numercal dsspaton s effectvely dspersed by physcal vscosty. Wth no localzed errors, the soluton s able to converge correctly down by fve orders. Acknowledgments Ths work was funded by the UK Engneerng and Physcal Scences Research Councl and the Defence Evaluaton Research Agency (DERA, UK Mnstry of Defence). The authors are ndebted to Trevor Brch of DERA for brngng the phenomenon of phantom vortcty to ther attenton and provdng nvaluable advce. [] Prnce SA. The Aerodynamcs of Hgh Speed Aeral Weapons. PhD Thess, Cranfeld Unversty, September 999. [3] Kwong CM, Myrng DF and Lvesey JL. Euler Calculatons of Flow Round a Mssle at Hgh Alpha. Departmental Report, Salford Unversty, Dec. 990. [4] Chnlov A. On Numercal Boundary Layers of Numercal Schemes Appled to Euler Equatons. Proceedngs of the Seventh Internatonal Symposum on Computatonal Flud Dynamcs, Bejng, pp 84-90, 997. [5] Turkel E. Accuracy of Schemes wth Nonunform Meshes for Compressble Flud Flows. Appled Numercal Mathematcs, North-Holland Publshng, pp 59-550, 996. [6] Batna JT. Implct Flux-Splt Schemes for Unsteady Aerodynamc Analyss Involvng Unstructured Meshes. AIAA Paper 90-0936-CP, 990. [7] Batna JT. A Fast Implct Upwnd Soluton Algorthm for Three Dmensonal Unstructured Dynamc Meshes. AIAA Paper 9-0447, 99. [8] Lou M-S and Hsu AT. A Tme Accurate Fnte Volume Hgh Resoluton Scheme for Three Dmensonal Naver-Stokes Equatons. AIAA Paper 89-994-CP, 989. [9] Qn N and Foster GW. Computatonal Study of Supersonc Lateral Jet Flow Interactons, J. Spacecraft and Rockets, Vol. 33, No 5, pp 65-656, 996. [0] Barbers D. Supersonc vortex flow around a mssle body. AGARD Advsory Report 303, Vol., Aug. 994. [] van Leer B. Upwnd-dfference Methods for Aerodynamc Problems Governed by the Euler Equatons. Lecture Notes n Appled Mathematcs, :37-336, 985. [] Anderson W, Thomas JL and van Leer B. Comparson of Fnte Volume Flux Vector Splttngs for the Euler Equatons. AIAA Journal 4:453-460, 986. References [] Marcon F. Concal Separated Flows wth Shock and Shed Vortcty. AIAA Journal, Vol. 5, No., Jan. 987, pp 73-75. 43.8