GRID CONVERGENCE ERROR ANALYSIS FOR MIXED-ORDER NUMERICAL SCHEMES

Transcription

1 GRID CONVERGENCE ERROR ANALYSIS FOR MIXED-ORDER NUMERICAL SCHEMES Cristoper J. Roy Sandia National Laboratories* P. O. Box 5800, MS 085 Albuquerque, NM AIAA Paper Abstract New developments are presented in te area of grid convergence error analysis and error estimation for mixed-order numerical scemes. A mixed-order sceme is defined ere as a numerical metod were te order of te local truncation error varies eiter spatially (e.g., at a sock wave) or for different terms in te governing equations (e.g., first-order convection wit second-order diffusion). Te case examined erein is te Mac 8 laminar flow of a perfect gas over a spere-cone geometry. Tis flowfield contains a strong bow sock wave were te formally second-order numerical sceme is reduced to first order via a flux limiting procedure. Te mixedorder error analysis metod allows for non-monotone beavior in te solutions variables as te mes is refined. Non-monotonicity in te local solution variables is sown to arise from a cancellation of first- and second-order error terms for te present case. Te proposed error estimator, wic is based on te mixed-order analysis, is sown to provide good estimates of te actual error. Furtermore, tis error estimator nearly always provides conservative error estimates, in te sense tat te actual error is less tan te error estimate, for te case examined. C D C f DE Nomenclature drag coefficient skin friction coefficient discretization error Senior Member of Tecnical Staff, cjroy@sandia.gov, Member AIAA * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockeed Martin Company, for te United States Department of Energy under Contract DE-AC04-94AL Tis material is a work of te U. S. Government and is not subject to copyrigt protection in te United States. F s factor of safety (F s =3) f solution variable g i i t order error term coefficient measure of grid spacing ( k =[N /N k ] / ) N number of mes cells p spatial order of accuracy q dynamic pressure, N/m R N nose radius (R N = m) St Stanton number (dimensionless eat transfer) r grid refinement factor u axial velocity component, m/s x axial coordinate, m y radial coordinate, m ε k+,k difference between a solution variable on mes k+ and mes k (ε k+,k = f k+ - f k ) γ ratio of specific eats (γ =.4) ρ density, kg/m 3 Subscripts and Superscripts exact exact value inf freestream value k mes level, (k =,, 3, etc., fine to coarse) n flowfield node index ~ estimated value to order p+ Introduction As computers become faster and algoritms become more efficient, computational fluid dynamics (CFD) as enormous potential to contribute to te design, analysis, and certification of engineering systems. However, simulation results are often regarded wit skepticism by te engineering community as a wole. Judging by te results of numerous blind validation studies, -3 tis lack of confidence in CFD is not surprising. To quote one autor, te results of suc exercises can be igly user-

2 dependent even wen te same CFD software wit te same models is being used. Oftentimes, wen a number of users do obtain te same results, tese results do not agree wit te experimental data. In order for CFD to acieve its potential, more work must be done to quantify te uncertainty in simulation results. Te uncertainty, or error, of a given CFD simulation can be categorized into two distinct areas. 4,5 Verification deals purely wit te matematics of a cosen set of equations, and can be tougt of as solving te equations rigt. Validation, on te oter and, entails a comparison to experimental data (i.e., te real world) and is concerned wit solving te rigt equations. Wit regards to te sequence, verification must be performed first for quantitative validation comparisons to be meaningful. Topics tat are included under te broad eading of verification include coding errors, incomplete iterative convergence error, truncation error, round-off error, far-field boundary error, and grid convergence (or discretization) error. Tis last source of error is related to te adequacy of te computational mes employed and is te focus of te current paper. For complex problems, te most reliable metods 4 for assessing te grid convergence errors in te solution to partial differential equations are a posteriori metods based on Ricardson Extrapolation. Roace as proposed a Grid Convergence Index 6 (GCI) as a uniform metod for reporting te results of grid refinement studies. As a minimum requirement for demonstrating solution accuracy, two grid solutions are used along wit a knowledge of te nominal order of accuracy of te numerical sceme to produce an error estimate in te solution properties. However, tis minimum requirement can be misleading wen te observed order of accuracy differs from te nominal order of accuracy. In Ref. 6, Roace furter promotes te idea of using an additional grid level (or levels) in order to verify te order of accuracy of te numerical metod and insure tat te solutions are in te asymptotic grid convergence range. Error analysis metods for ig-speed compressible flows can be complicated by te presence of sock discontinuities. Te most common numerical metods used for ig-speed flows are caracteristic-based upwind metods. For steady flows, metods tat are second order in space are often employed due to teir favorable mixture of accuracy and numerical stability. In order to prevent non-pysical oscillations, most upwind scemes employ limiters wic reduce te spatial accuracy to first order troug sock waves. In fact, Van Leer sowed tat te capturing of a discontinuity witout oscillation required tat te spatial accuracy of te sceme reduce to first order. 7 Te presence of bot second- and first-order spatial accuracy (at discontinuities) can greatly complicate grid convergence analyses. Carpenter and Casper 8 conducted a careful study of te grid convergence beavior for a two-dimensional ypersonic blunt-body flow. Teir study employed iger-order metods and omitted any flux limiting at te sock wave. Wile te numerical scemes tey employed were nominally tird and fourt order, tey found tat te spatial order of accuracy always reverted to first order on sufficiently refined meses. Teir findings indicate tat even witout te use of flux limiters to reduce te spatial order of accuracy at discontinuities, te information is passed troug te sock wave in a first-order manner (at least in two dimensions and iger). Similar results ave been observed by oter autors. 9- For sock-containing flows, it is surmised tat te local truncation error reduces to first order at te discontinuity, regardless of te use of flux limiters. Since te truncation error determines te order of te metod, tis spatial variation in te truncation error results in a mixed-order sceme. In addition to flows wit sock waves, tere are many oter examples of mixed-order numerical scemes. Leonard s QUICK sceme employs a tirdorder accurate convective operator and standard secondorder central differences for diffusion. Celik et al. 3,4 examined te subsonic, backward-facing step problem wit a numerical sceme wic used central differences for te diffusion terms, but was a mixture of first-order upwind and second-order central difference for te convective terms. A mixed-order beavior can also occur for cases were te transport properties undergo large, abrupt canges, suc as at te interface between two porous media. 4 Previous work by Roy et al.,5 verified te presence of bot first- and second-order errors for a ypersonic blunt-body flow wit a nominally second-order numerical sceme. It was sown tat te use of a mixed-order numerical sceme resulted in non-monotonic convergence of some of te flow properties as te mes was refined. Tis non-monotonic grid convergence beavior was found to occur wen te first- and second-order error terms were of opposite sign, tus leading to error cancellation. Non-monotonic grid convergence as been observed by a number of oter autors. For example, Celik and Karatekin 4 examined te flow over a backward facing step using te k-ε turbulence model wit wall functions. Tese autors found significant nonmonotonicity in bot te velocity and turbulent kinetic energy profiles as te grid was refined. Te two main goals of te current paper are to explore in detail te beavior of a mixed first- and secondorder numerical sceme as te grid is refined and to develop an error estimator wic can be applied to suc scemes. Te test problem used is te Mac 8 laminar flow of a perfect gas (γ =.4) over a spere-cone geom-

3 etry. Te accuracy of te surface pressure solutions on several grid levels was presented in Ref. and included a detailed model validation study. Tese same calculations are explored furter in te current paper, wit a focus on te analysis and estimation of grid convergence errors. Computational Model Te computational fluid dynamics code used erein is SACCARA, te Sandia Advanced Code for Compressible Aerotermodynamics Researc and Analysis. Te SACCARA code was developed from a parallel distributed memory version 6-9 of te INCA code, 0 originally written by Amtec Engineering. For te present simulations, te SACCARA code is used to solve te Navier-Stokes equations for conservation of mass, momentum, and energy in axisymmetric form. Te perfect gas assumption is made, and te flow is furter assumed to be laminar. Furter details of te flowfield models employed can be found in Ref.. Te governing equations are discretized using a cell-centered finite-volume approac. Te convective fluxes at te interface are calculated using te Steger-Warming flux vector splitting sceme. Second-order reconstructions of te interface fluxes are obtained via MUSCL extrapolation. Te viscous terms are discretized using central differences. A flux limiter is employed wic reduces to first order in regions of large second derivatives of pressure and temperature. Tis limiting is used to prevent oscillations in te flow properties at sock discontinuities. Te use of flux limiting results in a mixture of first- and second-order accuracy in space. Te implications of te mixed-order sceme on te convergence beavior of te metod as te grid is refined will be discussed in detail. Te SACCARA code employs a massively parallel distributed memory arcitecture based on multi-block structured grids. Te solver is a Lower-Upper Symmetric Gauss-Seidel sceme based on te works of Yoon et al.,3 and Peery and Imlay, 4 wic provides for excellent scalability up to tousands of processors. 5 Te simulations presented erein were run using a single 400 MHz processor of a Sun Enterprise 0000 saredmemory macine. Flowfield Conditions Te conditions used in te current simulations are presented below in Table. Tese conditions correspond to tose employed in te Joint Computational Experimental Aerodynamics Program (JCEAP) experiment conducted at Sandia National Laboratories by Oberkampf and Aescliman. Tis experimental data set consists of bot force and moment data 6 as well as ig-quality surface pressure data. 7,8 Since te focus of tis paper is on te numerical accuracy of te simulations, comparisons to experimental data are omitted. See Ref. for an extensive model validation study. For plotting purposes, te spatial coordinates are normalized by te nose radius (R N = m). Table Test conditions for JCEAP experiments Flow Parameter Freestream Mac Number Stagnation Pressure Stagnation Temperature Freestream Static Pressure Freestream Static Temperature Freestream Unit Reynolds Number Freestream Dynamic Pressure Wall Temperature Value N/m 63.8 K 86.8 N/m 47.7 K /m N/m Iterative Convergence Iterative convergence was assessed by monitoring te L norms of te residuals for te momentum equations. Since te flowfield is two-dimensional/axisymmetric, laminar, and as no flow separation or cemical reactions, te residuals were reduced down to macine zero (approximately fourteen orders of magnitude), tus insuring convergence of te iterative algoritm. Extrapolation Tecniques Te following development is based on a series expansion 4 of discretization error on mes level k DE k = f k f exact 36.7 K () were f k is a discrete solution value on mes level k and f exact is te exact solution. Eq. () may be applied on a point-by-point basis locally witin te domain or to global quantities (suc as lift and drag). For a uniform mes, tis series expansion may be written as 3 f k = f exact + g k + g k + g 3 k + O k 4 () 3

4 were g i is te it order error term coefficient and k is some measure of te grid spacing on mes k. For a second-order sceme, te g coefficient will be zero. Te general procedure is to write Eq. () for a number of different mes levels and solve for an approximation to f exact and te error term coefficients. In certain cases, a general error term of order p will be employed, were bot te coefficients and tis observed order p may be solved for. Witout loss of generality, te fine grid spacing is normalized to unity (i.e., =), and te grid refinement factor is defined as r k, k + = k + k (3) Some of te required assumptions for using tese extrapolation metods are tat te solutions must be in te asymptotic grid convergence range, te solutions must be smoot, and te local error sould be an indication of te global error. Some of te caveats for using tese extrapolation metods are tat tey tend to magnify roundoff and incomplete iterative convergence error, and tat te extrapolated solution generally does not obey te same conservation laws wic are obeyed by te original solutions. Standard Ricardson Extrapolation In te early 900 s, Ricardson 9,30 developed a metod of extrapolating two discrete second-order solutions to yield a fourt-order accurate solution. Te solutions were obtained on a fine grid wit spacing and a coarse grid wit spacing, wit / = (i.e., grid doubling/alving). Te fourt-order accuracy of te extrapolated solution arose from te use of central differences, wic contain only even powers in te expansion given in Eq. (). Unless central differences are used exclusively, te odd terms sould be included, tus for a secondorder numerical sceme, te two discrete solutions may be generally written as f = f exact + g + f = f exact + g + O 3 O 3 By neglecting te terms of order 3 and iger, te above system can be solved for approximations to f exact and g (te coefficient of te second order error term) g f f = f f = f (4) (5) were te overtilde (~) denotes approximate values wic neglect iger-order terms. Defining te difference in two successive grid levels as ε = f f and taking =, Eqs. (4) and (5) reduce to g = --ε 3 = f --ε 3 (6) (7) In general, te above relations for g and are tird-order accurate, owever tey will be fourt-order accurate wen central differences are used. Te assumption tat = and tat te odd error terms are present will be used for te remainder of tis section. Generalized Ricardson Extrapolation Te above Ricardson Extrapolation tecnique can be generalized to arbitrary grid refinement factor r and order p following Roace. 4 Te series representation is written as f = p f exact + g p + f = p f exact + g p + p + O ( ) p + O ( ) Approximating te above equations by dropping te iger-order terms and ten solving for te p t order error coefficient g p and te exact solution f exact results in g p ε = r p ε = f r p (8) (9) were r = r from Eq. (3). In tis case, te order of te discretization p must be assumed a priori since only two solutions are used. Te above estimates will in general be (p+) t order accurate. In addition to using te extrapolated values to estimate te errors in te discrete solutions, it is strongly recommended tat te order of accuracy also be verified. Tis type of order verification requires tree discrete solutions wic are monotonic as te grid is refined. Recovery of te formal order of accuracy of te sceme furter requires tat te tree grid solutions be in te asymptotic grid convergence range. Te series representation is now expressed as 4

5 p f = f exact + g p + p f = f exact + g p + p f 3 = f exact + g p 3 + p + O ( ) p + O ( ) p + O ( 3 ) (0) () () If te iger order terms are neglected, ten te above equations can be solved for approximations to te order p, g p, and f exact to give p r p p r r3 g p ε = ε + ε 3 ε = p r ε = f p r (3) (4) (5) Notice tat Eq. (3) is transcendental in p, and tus must be solved iteratively (see Ref. 4). For te case wen r = r 3 = r (i.e., constant grid refinement factor), tis equation reduces to p ln( ε 3 ε ) = ln( r) (6) Tis last equation was used by de Val Davis in Ref. 3 to solve for te observed order of accuracy for te natural convection in a square cavity. Wen te tree discrete solutions do not converge monotonically as te grid is refined, ten ε 3 /ε < 0 and Eq. (6) is terefore undefined. Celik and Karatekin 4 addressed te issue of non-monotone, or oscillatory, grid convergence by inserting a negative sign in front of te g p coefficient in Eq. () (te medium mes). However, matematical justification for suc a procedure is not well-founded. Tis paper will present bot an analysis metod and an error estimation tecnique for dealing wit non-monotone grid convergence. Mixed st + nd Order Extrapolation For te case wen bot first- and second-order error terms are included,,5 tree discrete solutions are needed, were te series representation is written as f = f exact + g + g + f = f exact + g + g + f 3 = f exact + g 3 + g 3 + O 3 O 3 O 3 3 Te inclusion of more tan one error term in te expansion is not a new concept. For example, see Ref. 3 for a discussion of Romberg interpolation as applied to te trapezoidal rule. For arbitrary mes refinement, te above equations may be solved for approximations to g, g, and f exact to yield g g ε 3 ( r ) + ε r ( r 3 ) = r ( r ) ( r 3 ) ( r r 3 ) ε 3 ( r ) ε r ( r 3 ) = r ( r ) ( r 3 ) ( r r 3 ) ε 3 ( r ) ε ( r r 3 r r 3 ) = f ( r ) ( r 3 ) ( r r 3 ) For example, if =, =.5, and 3 =, ten r = 3/ and r 3 = 4/3, and te above equations simply reduce to g = 5ε3 + 7ε g = ε 3 ε = f + 3ε 3 5ε For te case of constant grid refinement factor, te above equations simplify to g g r ε ε 3 = rr ( ) ε 3 rε = rr ( + ) ( r ) ε 3 ( r + r )ε = f ( r + ) ( r ) and for r =, tese equations furter reduce to (7) (8) (9) 5

6 Te above estimates are generally tird-order accurate. A tird-order error term could be easily included by simply adding anoter mes level (see Appendix A). Flowfield Grids Solutions were obtained for two different families of meses. Te first mes family contains six grid refinement levels, from Mes ( cells) to Mes 6 (5 5 cells), wit eac successive grid level found by eliminating every oter grid line in te two spatial dimensions (i.e., grid alving). Te second family of meses was obtained in te following fasion. A second-order accurate interpolation procedure was applied to a mes (not sown), resulting in a cell mes. Every tird point was retained from tis mes, resulting in a cell mes (Mes 0.5). Tis mes, altoug not used, is te baseline mes for te second family of meses (Meses.5 troug 6.5) wic are determined by again eliminating every oter grid line in bot directions. Te meses used in tis study are summarized below in Table. Mes Name g = --ε + ε 3 g = --ε --ε = f + --ε --ε Table Flowfield meses Mes Family a Mes Cells Grid Spacing, b Mes A Mes.5 B Mes A Mes.5 B Mes 3 A Mes 3.5 B Mes 4 A Mes 4.5 B Mes 5 A Mes 5.5 B Mes 6 A Mes 6.5 B a Family A as (3 5 n, n=0,,..., 5) cells in eac direction, wile Family B as ( 5 n, n=0,,..., 5) cells in eac direction b Te grid spacing measure is normalized by te grid spacing on te finest mes (e.g., Mes as =) Grid Convergence Error Analysis A contour plot of Mac number is presented in Fig. along wit te flowfield mes. Te mes is used for clarity. A strong sock wave occurs in te domain rougly alfway between te body and te outer boundary. Along te stagnation streamline (y/r N = 0), te sock is normal to te y-axis and is effectively grid aligned. For te remainder of te domain, te sock is generally not aligned wit te mes. In addition, no effort was made to cluster te grid to te sock. y/r N Mac Number Mes4(60x60Cells) Fig. Grid and Mac contours for JCEAP simulations. Te order of accuracy for te surface skin friction distributions as been calculated using Eq. (6) wit r = (see Fig. ). Recall tat te underlying assumption for tis equation is tat te solutions must cange monotonically as te grid is refined. Te results indicate tat te local order of accuracy varies from negative values to values as large as nine. Te undefined values, wic are not included in te figure, occur wen te argument of te natural logaritm in Eq. (6) is negative (i.e., te solutions are not monotonic). Te failure of Eq. (6) to provide an observed order of accuracy close to te nominal order of te sceme (second order) provides te motivation for te current paper. Te first step towards quantifying te observed order of accuracy of te metod is to examine te beavior of some norms of te spatial error. In order to calculate error norms, local estimates of te exact solution are required. Estimates of te exact solution were obtained by extrapolating te solutions using Meses,, and 3 (restricted onto a node mes) using te st + nd order extrapolation metod presented in Eqs. (7)-(9). Altoug not employed in te current work, Roace and Knupp 33 developed a metod for obtaining estimates of 6

7 Order of Accuracy, p Meses,, and 3 Skin Friction Coefficient two (grid alving) was used for eac grid family, te discrete solutions for Family A fall at,, 4, 8, 6, and 3, and te discrete solutions for Family B fall at.5, 3, 6,, 4, and 48. Te density norms exibit nearly firstorder beavior for all mes sizes. Te axial velocity exibits a region of second-order beavior wic asymptotes to a first-order slope on te finer meses. Te firstorder asymptotic beavior as te mes is refined is not unexpected, as discussed previously in te introduction. fexact st+nd Order Extrap Fig. Order of accuracy of te surface sear stress distributions from Eq. (6) using Meses,, and 3. te exact solution on te fine mes points. Te L and L norms were ten calculated as follows: L norm k L norm k N f k, n, n n = = N N f k, n, n n = = N were k indicates te mes level and n is summed over te N points used in te norm calculation. Due to te requirement of tree grids in te local calculation of te norms for mes Family A were calculated on eiter a 6 6 node mes or te grid for Mes k, wicever was smaller. For mes Family B, te norms were calculated for te smaller of a 4 4 node mes and te grid for Mes k. All of te norms employed only alf of te available flowfield points from te body out towards te sock. Te omission of te sock wave was required since te extrapolation tecnique used to approximate f exact is not valid for discontinuous solutions. Tese norms are presented in Fig. 3 for te mass density and te axial component of velocity. Te norms are all normalized to unity at te coarsest grid level (Mes 6.5) for convenience. Te norms are plotted versus te measure of te grid cell size on a log-log plot. Recall tat tis cell size was normalized suc tat = on te finest mes (Mes ). Since a grid refinement factor of Norm Te general first-order convergence beavior for te flowfield density norms could be a result of eiter te first-order flux limiter at te sock wave or an error in te code. In te latter case, it is important to note tat te code will produce te correct solutions based on prior validation work (see for example Ref. ), but may approac tese correct solutions at a less tan secondorder rate for certain flowfield properties. Tis reduction in order as been observed for relatively minor coding errors 34 (e.g., an incorrect array index, an incorrect constant in a difference operator, etc.). Te beavior of te surface eat transfer (Stanton number) wit mes refinement is presented in Fig. 4 at te axial location =.. For all but te coarsest mes, te Stanton number converges monotonically as te grid is refined (i.e., as 0). Te solid line represents an estimate of te exact solution found from Eq. (9) using Meses,, and 3. Furter insigt into te error beavior can be gained by examining te contributions of bot te first- and second-order error terms. Fig. 5 sows te beavior of te grid convergence error in te local Stanton number as te mes is refined. Te discrete solution error is calculated using te tirdorder accurate estimate for f exact from Eq. (9) (using L Norm of ρ L Norm of ρ L Norm of u L Norm of u st Order Slope nd Order Slope Fig. 3 Spatial error norms using alf of te points from te body to te outer boundary (excluding te sock). 7

8 =. Discrete Solution st + nd Order Extrap. predicts tat te coarse grid solutions will begin to exibit a second-order beavior. Te grid convergence analysis, wic uses only Meses,, and 3, qualitatively predicts te reduction in error exibited on te coarse meses. St =. Stanton Number Fig. 4 Beavior of te Stanton number as te mes is refined ( =.). Meses,, and 3) and te following relationsip: Discrete Solution st + nd Order Error st Order Error Term nd Order Error Term Spatial Error (%) f k (0) Te discrete error appears as te square symbols in Fig. 5 and is expected to be a good representation of te true error, especially for te coarser meses. Also sown in te figure are te normalized magnitudes of te first- and second-order error terms along wit te magnitude of teir sum: = g and g + g g () () Te first-order error term as a slope of unity on te log-log plot. For a first-order sceme, tis first-order error term will dominate te second-order error term, and te discrete solution error will coincide wit te first-order error. Te second-order error term as a slope of two, and will dominate te total error wen te sceme exibiting second-order beavior. Te magnitude of te sum of te two terms (solid line) is forced to pass troug te points associated wit Meses,, and 3 since tese solutions are used in te determination of te coefficients in Eqs. (7)-(9). First-order accuracy is seen in te fine grid solutions, wile te error analysis Fig. 5 Error in te Stanton number as te mes is refined ( =.). Te beavior of te skin friction coefficient as te mes is refined is presented in Fig. 6 for te same axial location. In tis case, te skin friction values first decrease as te grid is refined and ten increase at =. Te spatial error in te skin friction as te grid is refined is presented in Fig. 7. Te discrete skin friction values again exibit a second-order beavior on te coarser meses and a first-order beavior on te finer meses. Te coefficients of te first- and second-order error terms ( g and g ) are of opposite sign, tus giving error cancellation wen g = g. Tis error cancellation manifests as a sarp drop in te error predicted from using Meses,, and 3 (solid line). Tis error cancellation corresponds to te location were te discrete solutions cross over te estimated exact value in Fig. 6. As a result, te solution on Mes 3 (0 0 cells) is actually estimated to ave muc less error tan te finest mes (Mes wit cells). Te predicted error using Meses,, and 3 (solid line) agrees well wit te discrete solution errors sown in Fig. 7. Te estimated errors in te Stanton number at te stagnation point ( = 0) are given in Fig. 8. Te error analysis again predicts error cancellation between te first- and second-order error terms. First-order beavior is evident for te fine mes solutions, wile coarser solutions also appear to ave a second-order component. Te agreement between te predicted error and te discrete error is not as good in tis case. 8

9 0.00 =. Discrete Solution st + nd Order Extrap. 0 = 0 (Stagnation Point) Stanton Number Cf Discrete Solution st + nd Order Error st Order Error Term nd Order Error Term Fig. 6 Beavior of te skin friction coefficient as te mes is refined ( =.). Fig. 8 Error in te Stanton number as te mes is refined ( = 0). 0 0 =. Skin Friction Coefficient 0-4 = 0 (Stagnation Point) Skin Friction Coefficient Discrete Solution st + nd Order Error st Order Error Term nd Order Error Term 0-7 Discrete Solution st + nd Order Error st Order Error Term nd Order Error Term Fig. 7 Error in te skin friction coefficient as te mes is refined ( =.). Error estimates for te skin friction at te stagnation point are sown in Fig. 9. Since te exact solution for te sear stress at te stagnation point is zero, te error terms in Eqs. (0)-() are normalized by te freestream dynamic pressure (q inf = N/m ). Te error analysis predicts tat te error terms will be of te same sign and tus ave a smoot transition from second order on te coarse meses to first order as te mes is refined. In tis case, te first- and second-order coefficients ( g and g ) ave te same sign, so te magnitude of te sum of te error terms is larger tan eac of te individual error terms and te discrete solutions converge monotonically as te mes is refined. Fig. 9 Error in te skin friction coefficient as te mes is refined ( = 0). Te agreement between te predicted error and te discrete error is good for te finest meses. Tis error analysis as also been applied to te forebody drag, an integrated quantity. Fig. 0 gives te beavior of te drag coefficient as te mes is refined. A non-monotone beavior is seen on te tree coarsest meses only. Te spatial errors in te drag coefficient are presented in Fig.. Good agreement is again observed between te error analysis (using only Meses,, and 3) and te discrete solution error for te finer meses; owever, te non-monotonic beavior on te tree coarsest meses is not captured in te error analysis. 9

10 C D Drag Coefficient Discrete Solution st + nd Order Extrap. Fig. 0 Beavior of te forebody drag (excluding base drag) as te mes is refined Drag Coefficient Discrete Solution st + nd Order Error st Order Error Term nd Order Error Term Fig. Error in te forebody drag (excluding base drag) as te mes is refined. Grid Convergence Error Estimators In te previous section, te focus was on error analysis and, specifically, on understanding ow and wy non-monotonic beavior can occur for flow properties as te mes is refined. In tis section, te focus now sifts to error estimation. Te error estimation metods discussed below are intended to be used for engineering calculations were only a limited number of grid levels are available. In Ref. 6, Roace encourages te use of at least tree grid levels for problems sufficiently different tan tose previously studied. Tis position is also taken by te current autor. If te tree solutions converge monotonically as te mes is refined, ten te observed order of accuracy can be calculated wit Eq. (6), and te generalized Ricardson Extrapolation procedure of Eq. (5) can be employed using te two finest mes levels. If te solutions do not converge monotonically, ten te procedures developed in tis section are recommended. Te goal of tis section is to examine te beavior of a number of different error estimators for cases wen te flowfield properties converge non-monotonically. Te ideal error estimator would provide an error estimate tat is very close to te actual error and carries some statistical measure of te confidence tat te error estimate will be conservative (i.e., a σ or 95% confidence band). For complicated, nonlinear problems in multiple dimensions (and rarely in te true asymptotic grid convergence range), a rigorous proof of suc an error band is probably not attainable. 4,6 We are terefore forced to rely on more euristic metods of determining te uncertainties in CFD due to grid convergence errors. Te simplest metod would be to simply use te extrapolated estimate of f exact to estimate te error in te discrete solutions. For example, if te observed order of accuracy of some property as been verified (and te solutions are monotonic), ten te generalized Ricardson Extrapolation can be used to estimate te error from Eq. (0). However, since tere is an equal possibility tat te true exact solution is larger or smaller tan te estimated f exact, tis metod could be tougt of as producing a 50% confidence band (i.e., tere would be only a 50% cance tat te true error estimate would be smaller tan te estimated error). One approac would be to add a factor of safety to te error estimate of Eq. (0), suc as f Spatial Error (%) F k = s (3) were te factor of safety can be cosen as some appropriate value (e.g., F s = 3). Anoter common approac to reporting grid convergence studies is to report te difference between a coarse grid solution f and a fine grid solution f. Tese differences are generally reported as some percentage of te fine grid value. If te factor of safety is included, ten tis metod could be expressed as f Spatial Error (%) F f = s f (4) In Ref. 6, Roace points out te main problems wit tis approac, namely tat te error estimate is independent 0

11 of te order of accuracy of te numerical metod or te grid refinement factor used. Clearly te true errors would be quite different if a 5% error is found using Eq. (4) for bot a first-order sceme and a tird-order sceme. A similar statement can be made for two cases were te grid refinement factor was r = and r =.. Roace as proposed a uniform metod for reporting grid convergence studies wic properly accounts for te order of accuracy of te metod and te grid refinement factor. 6 Roace s Grid Convergence Index (GCI), on a percentage basis, is defined as GCI (%) F s f f = r p f (5) were p is te order of te sceme, r is te grid refinement factor, and te factor of safety is generally taken to be F s = 3. It can be sown tat Eqs. (3) and (5) produce similar error estimates wen te generalized Ricardson Extrapolation metod is used to determine in Eq. (3) and wen te GCI error estimate is less tan 0-0% (see Appendix B for details). As presented in Eq. (5), te GCI is a fine grid error estimator. In Ref. 6, Roace presents a simple extension of te GCI to be used as a coarse grid error estimator. Coarse grid error estimates can be useful wen a large number of parametric studies are required. In addition, all of tese error estimators must be normalized by some reference value (oter tan ) wen (or f for te GCI) approaces zero. Five different metods ave been used to estimate te grid convergence errors. Te first tree metods all employ Eq. (3), but differ in tat te estimate of te exact solution f is taken to be eiter te mixed st + nd exact order estimate of Eq. (9), or te generalized Ricardson Extrapolation value of Eq. (9) wit p = orp =. Also sown in te figure is Roace s GCI(%) assuming eiter p =orp =. Te mixed st + nd order metod required tree solutions to obtain te estimate of te exact solution. Coarser meses (from te same grid family) are used to provide te two additional solutions. For example, te st + nd order estimate at Mes 4 will also use te discrete solutions on Meses 5 and 6. Te oter estimators all require a single additional coarse grid, tus estimates are available on all but te coarsest meses of eac mes family. Tese estimates are compared to te best estimate error, wic is determined using te mixed st + nd order extrapolation metod on te tree finest mes solutions from grid Family A (Meses,, and 3) and omitting te factor of safety (i.e., F s = ). Tis best estimate error is expected to be a very good error estimate, especially on te coarser meses. Tese five error estimators are applied to te discrete values for te Stanton number at te =. axial location for te various mes levels, wit te results sown in Fig.. For tis case, all of te error estimation metods provide conservative estimates of te error over te entire range of grids. Te differences between te first-order extrapolation of Eq. (3) and te first order GCI(%) are negligible until te estimates get as large as 0% ( > 6). It sould be noted tat te first-order GCI and te second-order GCI will, by definition, differ only by a factor of tree (see Eq. (5)). Te error estimator results for te skin friction at =. are presented in Fig. 3. For tis case, all of te error estimates are conservative wit te exception of te second-order extrapolation (and second-order GCI) for = and =. In addition, te mixed st + nd order extrapolation gives error estimates muc closer to te actual (best estimate) error tan te first-order metods =. Stanton Number Best Estimate Error st + nd Order Extrap. st Order Extrap. nd Order Extrap. st Order GCI nd Order GCI Fig. Comparison of error estimates for te Stanton number ( =.). Error estimates for te Stanton number and te skin friction coefficient at te stagnation point are presented in Figs. 4 and 5, respectively. In bot cases, te second-order error estimates fail to provide conservative error estimates on a significant number of te meses. Tis figure igligts te dangers of simply employing te nominal order of te metod (in tis case second order) wen te sceme is actually of mixed order. For te Stanton number, te mixed-order metod provides conservative estimates of te error wic are consistently better tan te first-order estimates. For te skin friction coefficient, te first-order estimator and te mixed-order estimator give similar results except at =, were te latter metod is not conservative. It sould be noted tat tis is te only location were te mixed-order error es-

12 0 0 =. Skin Friction Coefficient 0-4 = 0 (Stagnation Point) Skin Friction Coefficient Best Estimate Error st + nd Order Extrap. st Order Extrap. nd Order Extrap. st Order GCI nd Order GCI Best Estimate Error st + nd Order Extrap. st Order Extrap. nd Order Extrap. st Order GCI nd Order GCI Fig. 3 Comparison of error estimates for te skin friction coefficient ( =.). Fig. 5 Comparison of error estimates for te skin friction coefficient ( = 0). 0 = 0 (Stagnation Point) Stanton Number Drag Coefficient 0 Best Estimate Error st + nd Order Extrap. st Order Extrap. nd Order Extrap. st Order GCI nd Order GCI Best Estimate Error st + nd Order Extrap. st Order Extrap. nd Order Extrap. st Order GCI nd Order GCI Fig. 4 Comparison of error estimates for te Stanton number ( = 0). timator fails to provide a conservative estimate of te error for all cases examined in tis paper. Error estimates for te forebody drag coefficient are presented in Fig. 6. Te mixed-order metod is te only metod tat provides conservative error estimates on all mes levels. Te first-order estimates, wile generally fairly good, do not give a conservative estimate of te error at =8. Anoter possible error estimator is to determine an observed order of accuracy by a weigting of te firstand second-order error terms from te mixed-order metod. Tese terms may be combined as Fig. 6 Comparison of error estimates for te forebody drag (excluding base drag). g p = g + g If a constant grid refinement factor is assumed as well as allowing = for convenience, ten te expressions for g and g from Eqs. (7) and (8) can be substituted into te above equation to produce p = + ϕ (6)

13 were ϕ = r ε 3 r ε r ε 3 rε 0 3 = 0 (Stagnation Point) Stanton Number Drag Coefficient Best Estimate Error p t Order Extrap. p t Order GCI Fig. 7 Comparison of p t order extrapolation error estimates for te forebody drag (excluding base drag). 0 0 Best Estimate Error p t Order Extrap. p t Order GCI Fig. 8 Comparison of p t order extrapolation error estimates for te Stanton number ( = 0). Tis observed order of accuracy can ten be used wit te generalized Ricardson Extrapolation procedure to obtain an estimate of f exact for use in Eq. (3). However, tis metod does not always produce conservative error estimates, as will be sown later. As an aside, tis metod for calculating te observed order of accuracy sould be used wit caution since it will always produce an order between one and two, regardless of te beavior of te discrete solutions. By no means can Eq. (6) be used to verify tat a code is indeed providing secondorder accurate solutions. Tis p t order extrapolation procedure as been applied to te drag coefficient results presented earlier and is sown in Fig. 7. Te estimates appear to be similar to te first-order error estimates sown previously in Fig. 6, and are not conservative for = 8. Also sown in te figure is te p t order GCI(%) using te standard metod for calculating te p from Eq. (6). Tis metod for calculating p is undefined for = 4, 6, and 8, and is tus not a useful error estimator for tis problem. Te p t order error estimators ave also been applied to te Stanton number at te stagnation point (see Fig. 8). Te p t order extrapolation fails te conservative test at =.5 and, wile te p t order GCI(%) is undefined for <3. Based on te above results, te best error estimation procedure for te current case is te mixed st + nd order metod. Tis metod almost always gave conservative estimates of te error, i.e., te error estimate was rarely smaller tan te actual error. In addition, tis metod generally provided error estimates tat were closer to te true (best estimate) error. Te mixed st + nd order error estimator is terefore recommended for mixed-order problems. Te first-order error estimator (along wit te first-order GCI) also provided fairly good results, but failed to give conservative error estimates for certain cases were te solutions were not monotonic wit grid refinement. Te mixed st + nd order extrapolation as been applied to te surface and field properties of te current simulations using Mes. Te error estimates for te surface eat flux (Stanton number) and sear stress (skin friction coefficient) are presented in Fig. 9 and 0, respectively. Te maximum errors in te eat flux occur at te stagnation point (6%) and at te sperecone tangency point (4%). Wit te exception of te stagnation region (were C f 0as 0, tus requiring normalization of te error estimates) te maximum errors in te sear stress are approximately 3.5% and occur near te spere-cone tangency point. Te relatively large errors at te spere-cone tangency point are due to te fact tis point is a surface curvature discontinuity. Altoug not employed in tis work, suc singular points sould be addressed eiter by additional grid refinement or oter special treatment. Te field errors in te mass density are sown in Fig. for te sperical nose region using Mes. Altoug not sown, te errors were by far te largest at te sock wave, probably due to te poor estimate of te exact solution in te presence of te sock discontinuity (recall tat te series expansion of Eq. () is not valid at 3

14 0 Error for Mes (480x480 Cells) Stanton Number 0 - y/r N Mes (480x480 Cells) Error in Density (%) Fig. 9 Error estimates along te surface for te Stanton number. Fig. Error estimates in te flowfield for te density. 0 Error for Mes (480x480 Cells) Skin Friction Coefficient Fig. 0 Error estimates along te surface for te skin friction coefficient. a discontinuity). Te large errors at te sock wave are partly due te misalignment of te bow sock and te grid lines. Errors appear to convect downstream from te sock wave. Te contour levels were scaled down to illustrate te beavior of te error between te sock and te body. Relatively large errors are also seen originating at te spere-cone tangency point. Large errors at tis location are expected since no effort was made to cluster te grid to tis discontinuity in surface curvature (second derivative of surface position). Te error appears to propagate from te spere-cone tangency point along te caracteristic Mac line. Summary and Conclusions Results were presented for te Mac 8 laminar flow of a perfect gas over a spere-cone geometry. A formally second-order numerical metod was employed; owever, te spatial order of accuracy of te metod was reduced to first order at te sock wave via a flux limiting procedure in order to prevent numerical oscillations. Te numerical sceme is terefore mixed-order in te sense tat te order of te local truncation error (wic determines te order of accuracy of te sceme) varies from second order over most of te domain to first order at te sock wave. Te first-order truncation error at te sock wave leads to te presence of a first-order discretization error component (owever small) everywere downstream. As te mes spacing is refined, tis firstorder error component eventually dominates te total discretization error. An error analysis metod was presented for mixedorder numerical scemes in wic bot first- and second-order error terms were included. Wen te coefficients of tese error components ave te same sign, te convergence of te solution properties as te mes is refined is monotone. However, wen tese coefficients are of opposite sign, error cancellation occurs at te crossover point were g = g, resulting in nonmonotonic beavior in te solution variables. Te proposed mixed-order error analysis captures te nonmonotone beavior of te solution variables, wereas metods based on linear extrapolation are unable to capture suc features. An error estimator was proposed wic used a 4

15 mixed-order extrapolation to obtain an estimate of te exact solution. Wen combined wit a factor of safety of tree (F s = 3), tis mixed-order error estimator was sown to provide good local estimates of te error, even on fairly coarse meses. Tis error estimator was furter sown to provide conservative estimates of te error in almost all cases examined, in te sense tat te true error was almost always smaller tan te error estimate. Error estimators based on a locally observed order of accuracy were also examined, but were sown to fail for certain cases wic exibited non-monotonic solution beavior wit grid refinement. Te error in te surface eat transfer (Stanton number) and surface sear stress (skin friction) was quantified using te mixed-order error estimator. Te largest errors in te eat transfer were found to occur at te stagnation point (6%) and at te spere-cone tangency point (4%). Te largest errors in te surface sear stress were approximately 3.5% and occurred near te sperecone tangency point. A field plot of te mixed-order error estimated in te mass density also sowed larger relative errors at te spere-cone tangency point. Tese errors appear to propagate downstream along a caracteristic Mac line. Te spere-cone tangency point is a surface curvature discontinuity and terefore requires grid clustering. Furter investigation into te grid convergence beavior of mixed-order numerical scemes is required. Te nature of te local reduction to first order for flows wit sock waves needs to be better understood. Examination of a control problem wic does not contain sock waves would also be elpful. Finally, oter sources of error wic can influence te discretization error sould be explored furter including: mes non-uniformities, boundary conditions, and singularities. Appendices Appendix A: Mixed st + nd + 3 rd Order Extrap. Altoug not employed in te current paper, it is possible to include first-, second-, and tird-order error terms in te analysis. In tis case, four grid levels must be used, and te series representation is f = 3 f exact + g + g + g 3 + f = 3 f exact + g + g + g 3 + f 3 = 3 f exact + g 3 + g 3 + g f 4 = 3 f exact + g 4 + g 4 + g O 4 O 4 O 3 4 O 4 4 Neglecting terms of order 4 and iger, te above set of equations can be solved for approximations to g, g, g 3, and f exact. Assuming tat te grid refinement factor is eld constant, te above equations reduce to g = ε 43 + r ( ε r 3 ε 3 ( r + ) ) r ( r + ) ( r ) g = ε 43 + rε 3 ( r + ) ε r r 3 ( r + ) ( r ) g 3 = ε 43 ε 3 rr ( + ) + ε r r 3 ( r + ) ( r ) ( r + r + ) = f + ε 43 + ε 3 ( r 3 + r + r ) ε ( r 5 + r 4 r r + ) ( r + ) ( r ) ( r + r + ) If te grid refinement factor is eld constant at r =, ten te above equations furter reduce to g g = -----ε 43 ε ε 3 5 = -----ε ε 3 --ε ε ε ε g 3 = 3 43 = f -----ε ε ε Te above equations are generally fourt-order accurate. Results for arbitrary grid refinement, altoug straigtforward, are somewat cumbersome and tus are omitted from te present work. Appendix B: Relation Between Error Estimators Wen te generalized Ricardson Extrapolation metod is used to estimate te exact solution f exact, ten te resulting error estimates, wen combined wit a factor of safety as in Eq. (3), can be sown to be approximately equivalent to Roace s Grid Convergence Index (GCI). Omitting te ( 00) factor for simplicity, te definition for te error of Eq. (3) for te fine grid may be written as f Spatial Error F = s (A.) Also recall te generalized Ricardson extrapolation formula from Eq. (9) 5