16th AIAA Computational Fluid Dynamics Conference June 2003 Orlando, Florida

Transcription

1 AIAA 3979 Algorithmic nhancements to the VLCAN Naier-Stokes Soler D. K. Litton, J.. dwards North Carolina State niersit, aleigh, NC J. A. White NASA Langle esearch Center, Hamton, VA 6th AIAA Comutational Fluid Dnamics Conference 3 6 June 3 Orlando, Florida For ermission to co or reublish, contact the coright owner named on the first age For AIAA-held coright, write to AIAA Permissions Deartment, 8 Aleander Bell Drie, Suite 5, eston, VA

2 AIAA 3979 Algorithmic nhancements to the VLCAN Naier-Stokes Soler D.K. Litton *, J.. dwards *, and J.A. White * Deartment of Mechanical and Aerosace ngineering Camus Bo 79 North Carolina State niersit aleigh, NC 7695 Hersonic Airbreathing Proulsion Branch Mail Sto 68 NASA Langle esearch Center Hamton, VA VLCAN (Viscous wind algorithm for Comle flow ANalsis) is a cell centered, finite olume code used to sole high seed flows related to hersonic ehicles. Two algorithms are resented for eanding the range of alications of the current Naier-Stokes soler imlemented in VLCAN. The first addition is a highl imlicit aroach that uses subiterations to enhance block to block connectiit between adjacent subdomains. The addition of this scheme allows more efficient solution of iscous flows on highl-stretched meshes. The second algorithm addresses the shortcomings associated with densit-based schemes b the addition of a time -deriatie reconditioning strateg. High seed, comressible flows are ticall soled with densit based schemes, which show a high leel of degradation in accurac and conergence at low Mach numbers ( M.). With the addition of reconditioning and associated modifications to the numerical discretization scheme, the eigenalues will scale with the local elocit, and the aboe roblems will be eliminated. With these additions, VLCAN now has imroed conergence behaior for multi-block, highl-stretched meshes and also can sole the Naier-Stokes equations for er low Mach numbers. Introduction The VLCAN (Viscous wind algorithm for Comle flow ANalsis) Naier-Stokes soler is considered the NASA standard in simulating reacting internal flows characteristic of high-seed roulsion deices. [] VLCAN can sole the Naier-Stokes or Parabolized Naier-Stokes equations using a ariet of discretizations, integration algorithms, turbulence models, and chemistr models and is alicable to general structured grids on multi-block domains. MPI message-assing is used to adat VLCAN to arallel architectures. The baseline integration strateg for Naier-Stokes alications within VLCAN is a diagonalized aroimate factorization scheme. This scheme is rather efficient on a er-iteration basis but is sensitie to the nearwall grid sacing. Conergence rates can degrade raidl for highl-stretched meshes. Furthermore, conergence rates can degrade when large numbers of blocks are used, due to the lack of strong couling between adjacent blocks. VLCAN is also designed for higher Mach number alications, and like most comressible flow solers, can eerience conergence degradation and solution inaccurac for er low Mach number flows. The resent effort is concerned with imroing the numerical efficienc of VLCAN for iscous flows on multi-block, highl -stretched meshes and for general low Mach number calculations. In the former contet, lanar relaation based imlicit methods are introduced, along with sub-iteratie rocedures that allow for a large degree of imlicit couling among blocks. In the latter contet, a time-deriatie reconditioning strateg based on the allseed flu formulae of [] is imlemented and tested for laminar and turbulent flows of caloricall- and thermallerfect gas at low Mach numbers. The cases used for alidation are the West and Korkegi double wedge of 4

3 AIAA 3979 of 4 configuration [3], a 3-D channel flow, iniscid flow oer a bum along a -D channel, a flat late simulation, and finall, a simulation of subsonic-tosuersonic flow transition in a two-dimensional nozzle. Goerning quations VLCAN soles the three-dimensional Naier- Stokes equations, eanded to include searate transort equations for indiidual secies and twoequation turbulence model comonents. Written in a generalized coordinate sstem, the Naier-Stokes set ma be written as, ) ( Q t dq () with the residual oerator ) (Q gien as ( ) ( ) ( ) S G G F F Q ) ( () The solution ector, Q, is gien b: ω κ t N w u Y Y J Q CS M (3) and the iniscid flu terms are defined as ( ) w u Y Y J t q NCS ω κ M ( ) V V V wv V uv V V Y Y V J F t z N CS ω κ M ( ) W W W ww W uw W W Y Y W J G t z N CS ω κ M (4) The Jacobian, J, of the transformation is defined as ( ) ( ) z J,,,, (5) The comonents of the cell face unit normal and the contraariant elocities are (6) z w u z w u V z w u W z (7) Definitions for iscous flu ectors,, F, and G and the source-term ectors can be found in []. In the aboe equations, Y i is the mass fraction of the i th chemical secies and N CS is the total number of chemical secies. For a caloricall erfect gas, the ressure, total enthal, static enthal, and total energ are gien b T ( ) w u h H C T h (8) P H t For a thermall erfect gas, the ressure, secies enthal, and static enthal are formed b the eressions: N CS i i i u Y T µ

4 AIAA 3979 T h h C dt (9) i i T h N CS i i h i Y i where i ( u /µ i ) is the i th secies gas constant. u is the uniersal gas constant and µ i is the secies molecular weight. Planar elaation Imlicit Flow Solers for Multi-Block Domains. Algorithm One of the major roblems encountered when soling three-dimensional roblems on large numbers of blocks is a reduction in the oerall conergence rate as the number of blocks increases. Tical domain-decomosition strategies used for finite-olume discretizations onl allow one or two mesh cells of oerla between adjacent domains. Tical imlicit solers, when formulated for multiblock arrangements, do not consider matri elements that would multil corrections generated in adjacent domains. Theore, subdomain couling is onl achieed in an elicit manner, through the residual ealuation at cells adjacent to block interfaces. The LX3D otion in VLCAN is built around a lanar relaation scheme for soling the sub-domain imlicit roblem, and is designed to sole the comlete (not arabolized) Naier-Stokes equations. The chosen swee direction ma be block-secific, and the crossflow lane linear sstem is aroimatel soled using an incomlete L decomosition rocedure. This aroach alleiates much of the numerical stiffness associated with highl-stretched mesh cells, roided that the crossflow lane is oriented so that to encomass the coordinate direction(s) with the largest degree of mesh stretching. Techniques such as Jacobian freezing are used to reduce the oerall CP load, and imlicit boundar conditions are incororated to further enhance stabilit. The LX3D otion has been tested for suersonic turbulent flow ast a clinder, laminar flow between two intersecting wedges [3] and -D unstead flow using a dual-timesteing aroach. An imroed imlicit algorithm has been deeloed with better block-to-block couling. M reresents the lanar relaation imlicit oerator as alied oer a subdomain, with its action uon a residual ector,, denoted b the oeration. M The actual Jacobian matri is denoted as A. Note that the factorization of M is defined onl oer the interior grid oints within a articular subdomain. In contrast, the Jacobian matri A contains elements that ma multil corrections that are obtained from the solution of the linear sstem in adjacent subdomains. Thus we ma slit the matri A into M N, where N contains elements of A that would multil corrections in adjacent subdomains and is the factorization error. Gien this, a general iteratie scheme for imroing the solution of the linear roblem at a articular subdomain ma be defined as: M ( Q for n, l Q n, l ) n A Q n, l n, l Q () Here, the inde n denotes a articular iteration leel for the solution of the nonlinear roblem (for unstead flows, this could be art of another subiteration), and the inde l denotes a articular iteration leel for the iteratie imroement of the solution of the linear roblem. With this basic strateg in lace, one can define an algorithm for imroing block-to-block couling: Sole: Q For l, l ma : M n, : Pass aroriate n n l Q, elements to ghost cells of adjacent blocks (arallel MPI send / receie) n, l n, l n n, l : Sole: Q Q M ( A Q ) nd loo date: Q Q Q n n n, lma This algorithm requires that an etra block diagonal matri, corresonding to the block diagonal of A, which is normall oer-written b the lanar IL factorization, be stored in addition to M itself. The onl change to the VLCAN inut deck necessar to imlement this algorithm is a flag indicating the number of subiterations erformed, l ma. If this is set to zero, then no subiterations are erformed and the lanar relaation scheme alone is used to adance the solution. Test cases shown in the esults section roide indications of the degree of imroement in conergence rates offered b this aroach. 3 of 4

5 AIAA 3979 Time-Deriatie Preconditioning Algorithm The time -deriatie reconditioning strateg currentl imlemented into VLCAN combines the rank-one reconditioning matri of Weiss and Smith [4] with the all-seed ersion of the low diffusion flu-slitting scheme (LDFSS) of dwards [5], deeloed according to a methodolog resented in dwards and Liou []. The reconditioning method can currentl be used with unge-kutta elicit time integration and diagonalized aroimate factorization imlicit time integration. The reconditioned Naier-Stokes equation set is gien b: dq P ( Q), () t where the reconditioning matri, P, is defined as r r T P I θ u () with r u T [ Y K YN u w H k ω ] r T P Q CS θ P Q P Q 3 V a K P Q Neq (3) where N eq reresents the total number of equations. The erence elocit, V, is resonsible for scaling the eigenalues of the equation set at low seeds to be of the same order. V is defined as min a r,ma V V, KV where a is the sound seed and V r, (4) is the elocit magnitude. V in the aboe equation acts as a cutoff elocit to reent singularities in the roimit of stagnation regions. In the VLCAN imlementation, the constant K scaling the cutoff elocit is a user inut, and V is set to the inutted free-stream elocit. The eigenalues of P are: Q,, [( M ) ± ( M ) 4V ] M V (4) As the incomressible limit is aroached, the eigenalues become a,, [ 4V ] ± (5) whereas the eigenalues reert to their traditional alues,,, and ± a as V? a. Numerical Discretization To ensure accurac at all flow seeds, it is necessar that the numerical discretization of the iniscid flu terms lect the reconditioned eigensstem. There are seeral aroaches for doing this, with the most rigorous being the use of characteristic analsis to derie reconditioned analogues of matri-dissiation methods. In the VLCAN imlementation, we instead imlement a reconditioned ariant of the low diffusion flu-slitting scheme (LDFSS) of dwards [5], deeloed according to a methodolog resented in dwards and Liou []. The interface flu in LDFSS is slit into conectie and I ressure contributions as follows: I C C [ C C ] [ D D ] a The ector L L C is the same as u r [ K ] z L L L (6) in q. (3), while (7) The reconditioned interface sound seed as a ( M ) ( M ) / 4V a is defined / (8) where the subscrit ½ reresents ealuation of the quantit using flowfield information arithmeticallaeraged to the cell interface. The quantities D, and D are functions of left-and right-state Mach numbers, seciall defined in terms of the interface sound seed and the erence Mach number as follows []: C, C, 4 of 4

6 AIAA 3979 [( M ) M ( M M ] ~ L, / L, / M ) (9) [( M ) M ( M M ] ~, /, / M ) () L with M L, a L / () For gas-dnamic flows, the use of the mo dified Mach number definitions in conjunction with the reconditioned sound seed enables the numerical dissiation mechanism of LDFSS to scale roerl at all seeds. cetions to this are the definitions for C and C, which contain a ressure-dissiation term roortional to L. As shown in [], this term acts to roide ressure-elocit couling at low seeds, and to ensure that this effect scales roerl, the term must be multilied b /,/ M. Precise definitions of the comonents of LDFSS ma be found in [5], and a more recent etension alid for general fluids ma be found in [6]. Time-Steing Scheme The solution is adanced in time b a reconditioned, diagonalized aroimate factorization (DAF) scheme. The reconditioned ersion of the DAF scheme is written as follows: T T T * n [ I J tsq ] Qi, j, k J tp ** * [ I J tδ ( )] λ, c λ, ( T ) Qi, j, k Q i, j, k *** ** [ I J tδ ( λ, c λ, )]( T ) Qi, j, k Qi, j, k *** [ I J tδ ( λ, c λ )]( T ) Q Q, In this, the modal matrices i, j, k n n Q i, j, k Qi, j, k Qi, j, k i, j, k () (3) T, ( T ), etc. are constructed from diagonalization transformations of the forms where T λ c T P, ( ) Q, (4) λ, c is a diagonal matri containing the eigenalues of the reconditioned uler sstem. Note that the first ste of the DAF rocedure is an aroimation of the more eact eression * n [ P J ts ] Q J t Q i j, k, (5) This aroimation allows the use of the Sherman- Morrison theorem to comute the action of P on the residual ector in O(n) oerations. The action of all other source Jacobian elements (those corresonding to chemistr and turbulence source terms) on the residual ector is comuted in a searate ste, inoling the use of Householder transformations to ensure good numerical stabilit. Other additions to VLCAN required for reconditioning include the use of characteristic inflow boundar conditions based on the reconditioned equations and the use of local time stes based on the eigenalues of the reconditioned sstem. esults Planar elaation esults The testing of the algorithm has been focused on the West-Korkegi intersecting-wedge geometr [3] and a channel-flow analogue formed b eliminating the wedges and the clustering to the leading edge. The clustering in the Y and Z directions remains the same, as do the length, width, and height of the (now) rectangular geometr. The free-stream conditions for the intersecting-wedge simulations are: M 3, e/m.e6, T 5 K. The free-stream conditions for the channel-flow simulations are: M.5, e/m 3.5e5, T 5 K. In both cases, the grid size is 6555 and laminar flow is assumed. nless otherwise mentioned, all cases were erformed in arallel on the North Carolina Suercomuting Center IBM -SP using a 6-block load-balanced decomosition of the baseline grid. Figure resents baseline results for the LX3D lanar relaation method ( l ma ). The ositie effect of using imlicit boundar conditions is clearl indicated, as is the fact that the lanar relaation rocedure allows a much higher CFL that the baseline DAF scheme. This translates in a significant CP saings, as the DAF scheme at a CFL of 3.5 takes.9 hours to run on the NCSC IBM-SP (6 rocessors) while the lanar relaation scheme at a CFL of 5 requires onl.95 hours. Figure resents results from a CFL-raming eercise erformed for the suersonic West-Korkegi case. The final CFL number is reached b raming from. to oer the first 5 iterations, from to 5 oer the net 5, and from 5 to the final alue oer the net 5 of 4

7 AIAA iterations. In this case and in most subsequent ones, the first 5 iterations are erformed on a coarse mesh using a first-order accurate iniscid flu discretization. Jacobian freezing is initiated after 5 iterations, with re-ealuation and factorization of the matrices erformed eer 5 iterations ast this oint. The controlling arameter l ma is set to one for this stud. Figure shows that there is little adantage to choosing a CFL higher than 5 for this case. Figure 3 shows the effect of the choice of swee direction on the erformance of the iteratiel imroed lanar relaation algorithm with l ma and l ma. Sweeing in the i direction (the direction of the dominant moement of the suersonic flow) is clearl erable to sweeing in the j direction. Performing one subiteration to imroe the solution of the linear sstem imroes the erformance in both cases, at least in terms of the number of iterations. Figure 4 illustrates the effect of aring l ma on the number of iterations required for conergence. As shown, the number of iterations required for conergence dros as the number of subiterations erformed increases. Also shown for comarison is a calculation erformed on the singleblock grid using l ma. This calculation was erformed on a Comaq S, which has enough shared memor to store all of the matri elements in core. Interestingl, the single-block grid erformance at l ma is slightl worse than the 6- block erformance. Otherwise, the trends are what one might eect. As the work increases significantl with the increase in the number of subiterations, it is instructie to eamine wall clock time. Figure 5 shows that for this redominatel suersonic flow, there is little benefit to erforming the subiterations, with onl about a 5% imroement in oerall eecution time for the best case of l ma. The net test case corresonds to Mach.5 flow through a channel similar in dimension to the West-Korkegi geometr. The single-block 6555 grid is decomosed into 6 blocks along the i coordinate. The CFL is ramed from a starting alue of. to a final alue of for the lanar relaation ariants, and again, 5 iterations are erformed to first-order satial accurac on the coarser mesh before interolating the solution to the finest mesh. Figure 6 ortras conergence histories for four runs: the first-order lanar relaation l l scheme with ma and ma, and the secondorder lanar relaation scheme with l ma and l ma. As shown, more than a three-fold imroement in the number of iterations required for conergence is eidenced for the first order discretization. CP times for the first order discretizations are 7.3 minutes for l ma and 33.4 minutes for ma. These times are also nearl a three-fold imroement. It is ossible that the sstem load ma hae been different for each of these runs, as the fact that the CP seedu is nearl in accord with the iteration count is somewhat surrising, gien the etra eense of the subiteration rocedure. It is noteworth that the use of subiterations stabilizes the second-order case to the oint that its conergence rate is er similar to the first-order case. Otherwise, as eidenced b the results, the calculation eentuall dierges. In comarison with the suersonic West and Korkegi case, these results indicate that the benefits of subiteratie imroement of the linear sstem solution ma be much larger for subsonic flows. The technique aears to aid in daming and/or eelling ressure disturbances that otherwise tend to lect from hsical and interface boundaries. Preconditioning esults The four test cases for the reconditioned sstem are flow oer a flat-late, flow through a two-dimensional TC nozzle, flow between intersecting wedges, and iniscid flow oer a bum in a channel. These corresond to ariations on standard test cases included in the VLCAN ackage. In all cases the maimum CFL is set to.5, and most cases inole both turbulent and laminar flow as well as multi-comonent gases. In all turbulent cases, the Wilco (998) k-ω model is used, while for all multi-comonent cases, a miture of nitrogen and ogen is used. Two-dimensional flow oer flat late Figure 7 shows the 659 grid for the flat-late simulation. The first run was to comare the results of raming the Mach number down from Mach.5 to Mach.5. The free-stream conditions for the Mach.5 simulation are: e/m.e7, T 3K. In each successie case the onl arameter changed was the Mach number, which decreased the enolds number b a factor of for each succeeding run. Figure 8 resents conergence histories for reconditioned and nonreconditioned cases at each Mach number. Strikingl, it can be seen that the conergence of the nonreconditioned sstem is not altered b lowering the Mach number, whereas the reconditioned sstem conerges faster for onl Mach.5 and.5. A closer inestigation into the solution roduced b the nonreconditioned sstem erified that the solution was in l 6 of 4

8 AIAA 3979 fact incorrect. This was erified when comaring with the Blasius solution at distances of.3,.4, and.45 meters from the front edge of the flat late. B using the Blasius solution for flow oer a flat late, the boundar laer thickness can be calculated as: 5. δ ( ) (6) e Table shows how the non-reconditioned sstem behaes in comarison to the reconditioned sstem. From this table it can be inferred that the nonreconditioned sstem is in fact conerging to an incorrect solution. The reconditioned sstem has a more realistic alue for the thickness of the boundar laer. For the Mach.5 run, the error for the reconditioned sstem is no larger than 6.7%. For the Mach.5 simulation, the error is less than.3%. Both of these numbers are in stark contrast to the 85-95% error found in the results obtained without reconditioning. Figures 9 and show the results for a turbulent, caloricall erfect and a turbulent, twosecies air flow oer a flat late, resectiel. The conergences for each case are er similar to the laminar flow in Figure 9. In all three eamles, the reconditioned Mach.5 case showed a marked imroement in conergence oer the nonreconditioned sstem. This is somewhat in contrast with the results from the eigenalue analsis, which indicate that the non-reconditioned sstem should hae a better oerall condition number at this Mach number. One reason ma be the resence, in the reconditioned flu-slitting, of ressure-diffusion terms that tend to smooth out ariations in the ressure field. As will be seen in the net few eamles, this result is consistentl indeendent of geometr. The conergence degradation indicated for the Mach.5 calculations could be associated with the decrease in enolds number. Stiffness due to low enolds numbers will not be alleiated b the inis cid reconditioning techniques currentl emloed in VLCAN. Two-dimensional flow through a nozzle The net case considered is flow through the two-dimensional TC nozzle. This is a standard test case for the VLCAN soler that inoles decomosing the nozzle flow into two regions: an ellitic region ustream of the nozzle throat and a arabolic region downstream of the throat. The ellitic region is soled using the DAF scheme (reconditioned and non-reconditioned), while the arabolic region is soled using sace-marching once the ellitic solution has been obtained. Figure shows the grid used in the calculation, while Figures and 3 show contours of Mach number and eddiscosit ratio (erenced to the laminar alue), resectiel. In the other calculations resented in this aer, the constant K scaling the cutoff elocit in q. (4) is set to one, since the elocit eerwhere is near the free-stream elocit. In the nozzle calculation, howeer, the free-stream sound seed is chosen as the erence elocit. Thus, a choice of K will not actiate reconditioning in the ellitic region. Figure 4 illustrates the effect of lowering K (equal to qctoff in the figure) on the conergence rates. The best results (for a four order-of-magnitude reduction) are obtained for K~.. Lower alues result in a significant degradation in conergence rate, though the sloe aears to be more consistent. These results indicate that the roer erence-elocit choice for strongl-mied subsonic / suersonic flowfields ma not be obious, and that trialand-error rocedures ma hae to be used to obtain the best results. en with this ambiguit, the use of reconditioning results in a factor of saings in iteration count. This translates into nearl a factor of saings in CP time, as the modifications necessar for imlementing reconditioning require er little additional CP time. Three-dimensional flow through intersecting wedges Figures 5-8 resent results from simulations of subsonic iscous flow through the West-Korkegi intersecting wedge geometr (shown in Figure 5) Freestream conditions are chosen to be the same as in the twodimensional flat-late case. Figure 6 illustrates the conergence histories for laminar flow through the wedges. The three non-reconditioned solutions do not conerge at the same rate and show worse conergences than the reconditioned scheme at all Mach numbers, ecet Mach.5. As the Mach number decreases in magnitude beond the Mach.5 case, the conergence rate is shown to decrease quickl. The reconditioned sstem contains a few oscillations but maintains its downward trend towards conergence. Figure 7 shows results for the turbulent, caloricall erfect case. This case shows trends that are almost identical to the laminar flow. The disarit in the results gien b the nonreconditioned sstem is magnified when running the two secies, turbulent simulation. As can be seen in Figure 8, the non-reconditioned residual at Mach.5 does not go down, but rather oscillates wildl around -. Although its reconditioned counterart does not disla this behaior, the conergence rate is noticeabl slowed down. The residual of the reconditioned sstem continues to go down towards conergence with minimal oscillations (in comarison to non-reconditioned sstem). 7 of 4

9 AIAA 3979 To erif that the results roduced b the reconditioning formulation were hsicall consistent, calculations of laminar flow through the channel analogue of the West-Korkegi geometr were also erformed. At locations far enough awa from the corner, it was anticiated that the boundar laer would deelo according to the Blasius scaling shown in q. (6). Table comares redictions from the reconditioned and non-reconditioned formulations ersus the Blasius result. The boundar laer thickness obtained from the nonreconditioned formulation turned out to hae an error of no less than 85%, while the reconditioned formulation roided results within a reasonable 6% of the theoretical alues. Conergence trends were similar to those corresonding to the intersecting wedges and are thus not shown. Iniscid flow oer a bum in a channel The last eamle of the alidation of the reconditioning strateg is iniscid flow oer a bum in a channel. For this articular case, the uler equations are soled, thus enolds-number effects illustrated in the earlier calculations are not resent. Figure 9 ortras the grid used for the calculations, while Figure shows the conergence histories for Mach numbers of.5,.5, and.5. These calculations reeal the eected trend of (nearl) Mach-number indeendent conergence rates when using the reconditioning technique. In contrast, the non-reconditioned formulation dislas a significant degradation in conergence rate as the Mach number is lowered. Finall, an imortant roert of the reconditioned DAF scheme is erified in Figure. The reconditioned scheme is designed to reert back to the comressible Naier Stokes equations for higher Mach-number flows. It is aarent from this figure that the two schemes are in fact identical for suersonic flow oer the bum, thus demonstrating the alidit of the reconditioned diagonalized aroimate factorization scheme for all flow seeds. Conclusions Two algorithms for enhancing the caabilities of the VLCAN Naier-Stokes soler hae been resented. The urose of the first algorithm is to imroe conergence rates for iscous flows on highl-stretched, mult i-block meshes. The algorithm decreases not onl the number of iterations to conergence, but also the time to conergence, an imortant factor in weighing the imortance of this new addition. So far, this addition is secialized for caloricall-erfect gases. The second algorithm is a timederiatie reconditioning strateg that is intended to eand the range of alicabilit of VLCAN toward low-seed, nearl incomressible flows. This addition is alid for caloricall and thermall-erfect gas and is designed for use with the baseline diagonalized aroimate factorization algorithm in VLCAN. Test cases show that the reconditioning framework greatl imroes solution accurac for low Mach numbers. Stiffness due to low-enolds number effects, is not, howeer, eliminated in the resent formulation, leading to some conergence degradation for low-seed, low enolds number flows. This ma require the use of iscous reconditioning strategies or a more imlicit treatment of the iscous terms. Future work will focus on combining the time-deriatie reconditioning techniques with the full-imlicit formulations to arrie at a framework caable of alleiating most sources of numerical stiffness resent in large-scale flow calculations. Acknowledgments This research was suorted b NASA Langle under grant NAG--5. IBM SP- comuter time was obtained from a grant from the North Carolina Suercomuting Center. eferences [] White, J.A. and Morrison, J.H., A Psuedo-Temoral Multi-Grid elaation Scheme for Soling the Parabolized Naier-Stokes quations, AIAA 9936, Jul, 999. [] dwards, J.. and Liou, M.-S. Low-Diffusion Flu- Slitting Methods for Flows at all Seeds, AIAA Journal, Vol. 36, No. 9, 998, [3] West, J.., and Korkegi,.H., Suersonic Interaction in the Corner of Intersecting Wedges at High enolds Numbers, AIAA Journal, Vol., No. 5, 97. [4]Weiss, J.M. and Smith, W.A., Preconditioning Alied to Variable and Constant Densit Flows, AIAA Journal, Vol. 33, 995,. 5. [5] dwards, J.. A Low-Diffusion Flu-Slitting Scheme for Naier-Stokes Calculations, Comuters & Fluids, Vol. 6, No. 6, 997, [6] dwards, J.., Towards nified CFD Simulations of eal Fluid Flows, (inited), AIAA -54CP,. 8 of 4

10 AIAA Planar relaation without imlicit BC's(CFL ) Planar relaation with imlicit BC's(CFL ) Planar relaation with imlicit BC's(CFL 5) Diagonalized AF (CFL 3.5) -.5 el L itsonthecoarse mesh; 5 itsonthefine mesh Jacobian freezing actiatedafter 5 itson each mesh Imlicit BC's inj and Kdirections el L CFL 5 CFL 5 CFL 65 CFL 5 CFL Figure : Conergence of lanar relaation scheme 5 5 Figure : ffect ofcfl number on conergence el L relaation swee in "i" direction; l ma relaation swee in "i" direction; l ma relaation swee in "j" direction; l ma relaation swee in "j" direction; l ma el L l ma ; single block grid l ma ; 6 block grid l ma ; 6 block grid l ma ; 6 block grid lma 3; 6 block grid l ma 4; 6 block grid Figure 3: ffect ofswee directiononconergence 3 Figure 4: ffect ofsubiteration number on conergence 9 of 4

11 AIAA 3979 IBM SP- wall clock time (seconds) l ma Figure 5: CP time required ersus number of subiterations el L lanar relaation, first order, l ma lanar relaation, first order, l ma lanar relaation, second order, l ma lanar relaation, second order, l ma 5 5 Figure 6: ffectof subiterations on subsonic channel-flow conergence LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach.5 5 el L Figure 7: Comutational Grid forflow oer a Flat-Plate Figure 8: Conergence histor for arious Mach numbers for laminar flow oer a Flat-Plate of 4

12 AIAA LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach LDFSS, Mach.5 P COND, Mach.5 LDFSS, Mach.5 P COND, Mach.5 LDFSS, Mach.5 P COND, Mach.5 el L el L Figure 9: Conergence histor for arious Mach numbers for a turbulent, caloricall erfect flow oer a Flat-Plate Figure : Conergence histor for arious Mach numbers for a turbulent, two-secies air flow oer a Flat-Plate of 4

13 AIAA 3979 el L noreconditioning -.75 qctoff.5 - qctoff qctoff.5 qctoff qctoff Figure 4:Conergencehistories: ellitic artoftcnozzlecalculation Z X Y LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach el L Figure 5: Comutational Gridfor West-Korkegi intersecting-wedges flow Figure 6: Conergence histor for arious Mach numbers forlaminar flow through West-Korkegi intersecting-wedges of 4

14 AIAA LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach LDFSS, Mach.5 P COND, Mach.5 LDFSS, Mach.5 P COND, Mach.5 LDFSS, Mach.5 P COND, Mach el L - el L Figure 7: Conergence histor for arious Mach numbers for a turbulent, caloricall erfect flow throughthe West-Korkegi intersecting-wedges Figure 8: Conergence histor for arious Mach numbers for a turbulent, two-secies air flow through the West-Korkegi intersecting-wedges LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach.5 LDFSS, Mach.5 PCOND, Mach el L Figure 9: Comutational Gridfor flow oer a Bum 3 4 Figure : Conergence histor for arious Mach numbers for flow oer a Bum 3 of 4

15 AIAA el L Non-Preconditioned Preconditioned Figure : ffect ofturbulent, suersonic flow on DAF calculation of channel flow Table : Boundar laer thickness for laminar flow oer Flat-Plate X- location d (meters) at Mach.5 d (meters) at Mach.5 (meters) Non-reconditioned Preconditioned Actual Non-reconditioned Preconditioned Actual % rror % rror % rror Table : Boundar laer thickness for laminar flow through Channel X- location d (meters) at Mach.5 d (meters) at Mach.5 (meters) Non-reconditioned Preconditioned Actual Non-reconditioned Preconditioned Actual % rror % rror % rror of 4