Carbuncle Phenomena and Other Shock Anomalies in Three Dimensions

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1 Carbuncle Phenomena and Other Shock Anomalies in Three Dimensions Keiichi Kitamura * and Eiji Shima Jaan Aerosace Exloration Agency (JAXA) Sagamihara Kanagawa 5-50 Jaan and Phili. oe University of ichigan Ann Arbor I 4809 USA Hyersonic flow comutations have roved to be very troublesome due to the aearance of shock anomalies (instabilities and oscillations) such as carbuncle henomenon. These anomalies are categorized into one-dimensional (D) and multidimensional (D) modes and these modes both arise from many factors and their combinations. Accurate rediction of hyersonic heating a key issue in hyersonic flow comutations is therefore challenging esecially for three dimensions (3D). In the resent study we focus on 3D shock anomalies and heating motivated by the following reasons: ) Intuitively D shock anomalies are considered to develo more likely in 3D than in two dimensions (D) but it cannot be roved mathematically nor has it been numerically demonstrated; secifically it is not clear yet whether the third dimension lays another role which is absent in D. ) ost of roosed remedies for D anomalies had been tested in D or D setus in the literature but it is not guaranteed whether such D dissiations actually work well in 3D. 3) It is already known to be troublesome to extend some of D methods develoed in D considerations to 3D. The numerical results show that 3D anomalies are too comlicated to be redicted from their D counterarts and that they can either be artly removed or (even worse) enhanced by D dissiations. Therefore robustness of a numerical method which worked well in D may not be reserved in 3D. * esearcher JAXA s Engineering Digital Innovation (JEDI) Center 3-- Yoshinodai Chuo ember AIAA. Senior esearcher and Director JAXA s Engineering Digital Innovation (JEDI) Center 3-- Yoshinodai Chuo ember AIAA. Professor Deartment of Aerosace Engineering Fellow AIAA.

2 Nomenclature c C E = secific heat at constant ressure = ressure coefficient = total energy E k F k = inviscid and viscous flux vectors in k-direction (k = 3 corresonding to x y z resectively) H = total enthaly = ach number = ressure Pr = Prandtl number 0.7 q e T V = heat-transfer rate = adius of shere 0. m = eynolds number = density = temerature = velocity u v w = velocity comonents in Cartesian coordinates x y z = Cartesian coordinates min = angle from the nose (where 0) of cylinder or shere = minimum grid sacing (near the wall) γ = secific heat ratio.4 = thermal conductivity = c /Pr = molecular viscosity Subscrits cell F- w = value based on the minimum grid sacing = Fay-iddell s redicted value = value on the wall = freestream value

3 0 = stagnation value H I. Introduction YPESONIC flow comutations have roved to be very troublesome due to aearance of shock anomalies (instabilities or oscillations) such as carbuncle henomenon []:(Figure for the case in tyical twodimensional (D) domain and Fig. for D normal shock). Although we are aware that classifications of such anomalous solutions are scattered among the researchers [-6] we show in Table what we currently believe to be the most convincing idea in which the carbuncles are categorized into multi-dimensional (D) instabilities. At any rate all those anomalies are known to arise from the following factors and their combinations [7-9]: flow conditions (ach number eynolds number and the ratio of secific heats) mesh (size asect ratio etc.) and numerical methods (flux function accuracy etc.). In articular the authors [9] recently reorted that any flux functions can lead to those anomalous solutions deending on the shock location relative to grid lines. oreover they made clear that there are at least two causes of the shock anomalies: one of these is a one-dimensional (D) effect and the other is D. The former aeared to be alleviated by adding (D) dissiation to the shock-normal direction; whereas the latter could usually be suressed by D dissiation in the shock-erendicular (transverse) direction [0-8]. However when both of the two causes arise at the same time these dissiations do not work well. Thus a flux function which is free from those two kinds of anomalies is needed. We do of course have several flux functions that can be tuned by an exerienced user to solve many secific roblems. In ractice the devil you know may be referable to the one that you don t. Accurate rediction of hyersonic heating a key issue in hyersonic flow comutations is therefore challenging [9 0] esecially for three-dimensional comlex geometries. For heating comutations the authors [] suggested the use of flux functions satisfying the following three roerties: I. Shock stability/robustness (i.e. free from both D and D anomalies) II. III. Conservation of total enthaly (and hence total temerature) An ability of the Euler solver to sharly resolve contact discontinuities. This is necessary if the associated Navier-Stokes solver is to resolving boundary-layers (and hence temerature gradients) economically. It turned out that unfortunately we had no flux erfectly satisfying all the roerties. Nevertheless the criteria introduced therein for hyersonic heating comutations and the classification of Euler fluxes are considered useful ieces of information in choosing/develoing Euler fluxes (for details lease see []).

4 In our revious work [9 ] we focused on D and two-dimensional (D) issues; in the resent study we will extend these discussions to three dimensions (3D) motivated by the following reasons:. Intuitively D shock anomalies are considered to develo more likely in 3D but it cannot be roved mathematically nor has it been numerically demonstrated. Secifically it is not still clear whether the third dimension (added dimension into a D setu) introduces behavior that is absent in D.. ost of roosed remedies [0-4] for shock anomalies had been tested in D and D setus in the literature with the claim that 3D extension is straightforward. It is true in a mathematical sense that the concet can be generalized but taking into account the revious observation it is hard to guarantee continued success. 3. It is already known to be troublesome to extend some of D methods develoed from D considerations to 3D. Yoon et al. [] for instance stated in their recent work that the difficulties in 3D extension encountered by their D-based limiter stem from the fact that cells do not belong to the same lane. Balsara [3] argued that one form of the HE iemann solvers is too dissiative in D while a simle extension of the original D flux to multi-dimensions limits Courant number and only works for Euler equations. oreover from the authors exerience a D hybrid flux using two vectors of normal and arallel to the shock [] can face a difficulty in determining the two shock-arallel directions in 3D sace even though one of those vectors can be (arbitrary) defined: The roblem was that the needed dissiation to suress shock anomalies may differ from one direction to another. We will extend and conduct our revious numerical exeriments [9 ] along with benchmark tests in [0] and htt://fun3d.larc.nasa.gov/chater-9.html#hyersonic_benchmarks retrieved on Jun in 3D for oular Euler fluxes. As in our revious work we took great care to eliminate any asymmetry from the grid or the initial data and no artificial erturbations were introduced in an attemt to trigger instability. Therefore we believe that all of the anomalies that we observe arise initially as comutational instabilities driven by rounding error. This accounts for the fact that many henomena took thousands of iteration to become visible. In ractice they might arise much earlier in resonse to non-smoothness of the data. Finally we will try to summarize 3D shock anomalies stressing both the similarities and differences with their D counterarts. II. Comutational ethod A. Governing Equations The governing equations are the comressible Euler or Navier-Stokes equations:

5 Q E k 0 : Euler t x k (a) E t x Q k k : Navier-Stokes k F x k (b) u k 0 Q ui Ek uiuk ik Fk ik T E u k H u j jk xk () u u u j k l jk jk xk x (3) j xl where is density u i velocity comonents in Cartesian coordinates E total energy ressure H total enthaly (H = E + (/)) and T temerature. The working gas is assumed to be air aroximated by the calorically erfect gas model with the secific heat ratio =.4. The Prandtl number is Pr=0.7. The viscosity is calculated by the Sutherland s formula and the Stokes hyothesis is emloyed: that is =-/3. B. Comutational ethod The following methods are used for comutations herein if not mentioned otherwise. As for satial discretization the rimitive variables at each cell-interface are simly interolated from the cellcenter values (first-order) for inviscid cases (since shock anomalies tend to develo more likely in first-order rather than in second-order [9]) (Cases # and # exlained later) or to second-order accuracy by Van Albada-limited [4] USC reconstruction [5] for viscous cases (Cases #3 and #4 again exlained later). The Van Albada limiter one of the most commonly used limiters is selected because of its better convergence erformance in general. Then inviscid fluxes at the cell-interface are calculated from oe (E-Fix) (oe [6] with Harten s entroy-fix [7]) Van eer s flux-vector-slitting (FVS) [8] AUS+ [9] or AUSPW+ []. oe s flux-difference-slitting (FDS) (reresenting FDS schemes in Grou ) has low dissiation although it is known to be vulnerable to shock anomalies (e.g. carbuncle henomenon) [7 9]. Van eer s FVS (reresenting FVS schemes in Grou ) on the other hand is known to be almost free from such shock anomalies but actually exhibits them in extreme cases []. AUS+ (Grou 3) which can be regarded as a mixture of FDS and FVS and so reresents Grou 3 is more stable than oe s flux [ 3] and reserves constant enthaly in steady flows although it also suffers from shock

6 anomalies under certain conditions [9 ]. AUSPW+ (Grou 3) is an imroved AUS+ equied with a multidimensional dissiation term. These fluxes were categorized as in Table []. Other D fluxes [-5] are also of interest but in this aer we will focus on only reresentative fluxes chosen from each Grou. However it should be admitted that our grous may not be ideally chosen. Hänel [30] a variant of Van eer s FVS having total enthaly conserving roerty are omitted since it reortedly [] behaves in almost the same manner as Van eer s FVS. The selected fluxes are briefly described in D forms below. oe (E-Fix) [6 7]: F F F Λˆ ΔQ The (^) stands for oe-averaged values. and are right and left eigenvectors resectively and (4) diagonal matrix of characteristic seeds with entroy-fix.5 0. if

7 Van eer s FVS [8]: otherwise v c u v c u c u c if sign if sign T 4 F F F (5) AUS+ [9]: P P a P P Φ F T T H v u P Φ otherwise if 4 otherwise if sign P 4 where u a a a a a a max ~ ~ ~ min * * (6a) (6b) (6c)

8 Viscous fluxes are comuted by using second-order central difference; while for time integration first-order Euler exlicit method (for inviscid cases) or U-SGS (for viscous cases) is emloyed. No turbulence model or real gas model has been used. Detailed information of the solver with regard to formulations and discretizations is found in [3]. III. Two- and Three-Dimensional Tests for Hyersonic Shock Anomalies In this section we erform and comare D and 3D numerical tests including a few reviewed results [9 ] that are essential to the resent discussions. As noted in the Introduction care was taken in all cases to ensure that the grids and initial data were symmetrical within rounding error. Euler equations are solved by a finite volume code first order both in time and sace. All the test cases and results are summarized in Table 3. A. Test #: Planar Shock D [9 ] These cases were already conducted extensively in [9 ] but here we extract only limited results that are essential to the resent discussions. AUSPW+ []: P P c c P P Φ Φ F 0 0 T T H v u P Φ If 0 m f w f f w otherwise f f w f w where 3 min w otherwise f S S 0 0 min min min and S P P (7a) (7b) (7c) (7d) (7e) (7f)

9 - Test #A: The grid consists of 50 5 square cells as shown in Fig. 3a. The numerical conditions e.g. = stes with CF=0.5 are the same in [9]. The initial shock osition arameter (see [9] or the descrition in Section III-B (you alternate between and ) is taken as 0.0 or 0.5 (the initial shock is imosed exactly on a cellinterface when =0.0 and at the cell-center when =0.5; see Fig. 3a). Tyical results are shown in Fig.. The examle result by using Van eer s FVS is shown in Fig. 3b and exhibits no evidence of shock anomalies. The results from other fluxes are summarized in Table 3. As shown in ef. [9] the mechanisms of multidimensional (D) shock anomalies are distinct from their D counterart although the former anomalies are related to the latter: D shock oscillation aears deending on the relative ositioning of the shock to the grid line; while the D oscillations can be triggered by D oscillations in certain conditions. Detailed exlanations for D and D anomalies are found in efs. [9 ]. - Test #B: The grid was extended to cells as shown in Fig. 4a. The comutations were conducted stes with the other conditions remained []. The D shock irregularities can develo if the numbers of grid oints is increased in the shock erendicular direction [] as shown in Fig. 4b. This suggested the ossibility that the increment of the grid oints is related to increment of degrees of freedom for numerical errors to develo. 3D - Test #C: The grid consists of 0 of the D grids (Test #A Fig. 3a) stacked in the additional dimension ( cells Fig. 5d). - Test #D: The grid consists of 0 of the D grids (Test #B Fig. 4a) stacked in the additional dimension ( cells). All the cells are isotroic (meaning that the same distances are maintained between grid oints along each mesh line) and the comutational conditions are the same as the D case. Only selected results will be resented below. The results for Test #C are shown in Figs The oe (E-Fix) (Fig. 5b) showed total breakdown (carbuncle) in three dimensions. The shock shae is irregular i.e. the instability occurs in every direction. esults of Van eer (Fig. 6) on the other hand are stable as in the D case (Fig. 3b). AUS+ (Fig. 7) showed a stable result for =0.0 but oscillatory for =0.5 in consistent with the D cases in ef. []. AUSPW+ (=0.0) initially showed regular oscillations in both directions (every bum has the same distance to each other in contrast to Fig. 5b) erendicular to the catured shock (Fig. 8a) but later the original lanar shae was recovered (Figs. 8b and 8c). This recovery

10 seems to be due to D dissiation term in AUSPW+. The similar behavior was already observed in D and in another D flux [] but its effectiveness in 3D has been confirmed at least in the current articular case. Test #D results are resented in Figs. 9 (Van eer) and 0 (AUSPW+). Astonishingly the 3D Van eer results in Fig. 9 are more stable than the D ones in Fig. 4b but in conjunction with the above hyothesis and the results in Figs. 3b and 4b numerical dissiation added by cells in the third direction seemed to have a favorable effect in this case. In AUSPW+ case (Fig. 0) in contrast reached an unstable solution: small random wiggles aeared (that is no regular attern is observed) at 5000 stes (Figs. 0a and 0b) the breakdown of the shock shae occurred (during stes Fig. 0c) and finally instability develoed and remained at stes (Fig. 0d). Note from Table 3 that the final state is similar to the corresonding D results rather than another 3D result in Test #C (Fig. 8c). Therefore the D dissiation term in AUSPW+ does not seem to be effective in cure of this D dominant anomaly. B. Test #: Hyersonic Flow over Blunt Body (Circular Cylinder) D [9] As in Test # in D these cases are reviewed only for reference. The (shock-aligned) grid used in this test had originally been rovided by Dr. Jeffery White et al. NASA angley. This grid was constructed by identifying one grid line with the shock roduced by an accurate shock-fitted solution and is rather surrisingly a very difficult grid on which to cature a shock. After removing some small asymmetries from the grid that we received we dilated the grid slightly so that the shock osition now lay in general between two grid lines. In revious work we made use of various dilations but here we used only two. When the arameter =0 we have the original grid as sulied and when =0.5 the shock should sit between two grid lines. The grid size and the comutational conditions are: - Grid: 0 (circumferential) 48 (wall-normal) - CF = Comutational timestes: stes - Flow condition: =6.0 The grid and examle result are shown in Figs. a and 3a. In the results D or (D-triggered) D shock anomaly (these terminologies have been categorized in Table ; see ef [] for more detailed exlanations) aeared on some grids whereas it did not emerge on the other grids (Fig. 5 and Table ). From this difference it Private communication with Jeffery White et al. NASA angley esearch Center Ar. 007.

11 had been revealed that the relative ositioning of the grid line to the shock layed an imortant role although this asect of shock anomalies is not discussed in this section but later in the next section. 3D The grid in D has been extended to three dimensions (00 cells in the third direction). - Grid: (evenly saced) The symmetry (reflection) condition is imosed at the sanwise boundaries and the other comutational conditions are the same as D cases. The results are shown in Figs. -3. Figure is focused on the develoment of the 3D carbuncle for oe (E- Fix) case. It is seen from the results that D shock anomalies develoed in every direction: ore recisely the carbuncle instability seemed to have occurred in the D slice (500 stes Fig. b). By 500 timestes the carbuncle has reduced but a sanwise oscillation has develoed (Fig. c). Then both of these anomalies combined (i.e. D carbuncle and 3D oscillations) until 5000 stes (Figs. d-f) and this catastrohic solution remained unchanged to stes (Fig. g) leaving bums randomly laced on the shock surface but showing the density residual decreased by more than seven orders of magnitude (Fig. a). Solutions of this kind i.e. converged carbuncles are also reorted in D (Fig. Fig. a or efs. [9 ]). In contrast to the lanar shock case (Fig. 5) the shock anomalies in this case seem to have develoed in different modes in different directions and also in different rates. These differences seem to be due to the fact that the cells are totally the same from one slice to another while within a slice the adjacent cells are different. Thus there aeared to be two ossibilities: i) The D carbuncle triggered 3D oscillations i.e. the aearance of the 3D oscillations was totally deendent on the D carbuncle. ii) The D carbuncle develoed faster than 3D oscillations i.e. the 3D oscillation gently and subliminally develoed while the D instability emerged. We examined this by introducing and comaring norms of velocity asymmetry defined as v = v uer - (- v lower ) = v(i j k) + v(i max +-i j k) (8) which is the measure of velocity difference between cells sharing the same x and z coordinates but having oosite signs in y with a similar definition for w (in the third dimension). According to Fig. the instabilities both in y and z directions arose from the very beginning of the comutation at the level of round-off errors and they grew

12 exonentially with time. Thus it is difficult to conclude which of the above hyotheses was right but at least there is the ossibility that D and 3D shock instabilities share the same cause and that they develoed in the same rate yet in different aearances. In AUS+ or Van eer results (Figs. 3b and 3c) however no evidence of shock anomalies is seen: These flow atterns remained stable and symmetric even in the 3D setu. They continued to do so even when the third dimension was extended to give a grid size of though the results are omitted. IV. Two- and Three-Dimensional Hyersonic Heating Tests for Navier-Stokes Codes In this section we carry out D and 3D viscous cases. Again no initial erturbation is introduced. Navier-Stokes equations are solved by a satially second order finite volume code with U-SGS imlicit time integration and the results are included in Table 3. A. Test#3: AUA Benchmark Hyersonic Heating Test [5] D This is a viscous hyersonic ( =7) benchmark test used for AUA (angley Aerothermodynamic Uwind elaxation. Algorithm develoed at NASA angley for hyersonic flow simulations) and FUN3D codes [5] emloying a shock-aligned grid. The grid was rovided by Dr. Peter Gnoffo NASA angley ** as a 3D mesh but one slice of it was taken and used for the D cases. Then as in the Test # we made the same modification on this grid i.e. the original grid ( =0.0) was dilated a half-cell width as the modified grid ( =0.5). The comutational conditions are given as follows. - Grid: min =.66e-6 m - CF = 00 - Comutational timestes: stes - Flow conditions: V = 5000 m/s ( =7) e = /m = 0.00 kg/m 3 T = 00 K T w = 500 K With the above setu the cell eynolds number is e cell =.00 ratio of Pitot ressure to free stream ressure P 0 /P = and Fay-iddell s [3] stagnation heating 46.5 W/cm (slightly smaller than AUA-redicted value [5] of 5 W/cm ). Comuted results are summarized in Figs One can see that oe (E-Fix) suffered from shock anomaly in the modified grid (Fig. 5a) (although the density residual droed to O(-6)) but not in the original grid (Fig. 4a). ** Private communication with Peter Gnoffo NASA angley esearch Center ar. 009.

13 This anomaly clearly affected surface ressure and more severely surface heating rates (more than 0% overestimation) (Figs. 6a 6b 7a and 7b). Other flux functions yielded symmetric and stable (at least eight orders dro in -norm of density residuals) solutions with excetion of AUS+ (oscillatory results on either of the two grids with only two orders reduction in residual). Van eer resulted in severely oor redictions of heating as exected. In addition we oint out here that a non-shock aligned grid roduced similar results as shown in []. 3D This is the 3D case of the viscous =7 hyersonic heating test which was reviously used for examle in ef. [5]. The grid was rovided by Dr. Peter Gnoffo as stated before. The grid system has cells with the symmetry condition at the sanwise boundaries and the rest of the flow and comutational conditions are the same as in the D test. The results for =0.0 are shown in Figs. 8. According to these figures the oe (E-Fix) case was affected by three-dimensional effect: the shock shae (Fig. 8a) surface ressure (about 5% Fig. 8b) and heating (more than 50% Fig. 8c) exhibited asymmetry in the crossflow direction (= the third dimension) as in Test #. It is noteworthy that in this case the oe (E-Fix) flux suffered from shock anomaly only in the third dimension and maintained the D stable symmetric solution (Fig. 4a) in each D lane: This is in contrast to the examle seen in Fig. (Test #) in which anomalies aeared in every direction. This seems to show that sometimes an anomalous mode is suressed by adding an extra dimension. Other fluxes showed almost as the same trends as in D cases: Van eer reached converged [to machine zero O(-6)] symmetric solution with surface heating underestimated; AUS+ again suffered from numerical oscillations (only two orders residual dro); AUSPW+ solutions are indistinguishable from the D solution. The results for =0.5 are similar [only with one excetion in oe (E-Fix) case which eventually blew u due to more severe oscillations] and hence shown only in Table 3. B. Test #4: Challenge Problem for Shere [33] 3D Although this roblem is axisymmetric it is solved on a fully 3D grid and is therefore a genuinely threedimensional test referred to as a challenge roblem in [33] of viscous hyersonic ( =) heating. A shock-

14 aligned grid rovided by Dr. Peter Gnoffo (through Dr. Bil Kleb) was used. We conducted this test to demonstrate how difficult it is to obtain satisfactory heating rofiles by existing methods in 3D and how much D terms work to imrove the solutions. - Grid: ( =) cells + ( =) cells = cells (Fig. 3a) - adius: = 0.m - CF = 00 - Comutational timestes: 0000 stes - Flow conditions: V = 467 m/s ( =) e = 0.5e+6 (based on radius ) e cell =5 (based on minimum grid sacing min =.e-6 m) = 0.06 kg/m 3 T = 300 K T w = 800 K With the above setu ratio of Pitot ressure to free stream ressure is P 0 /P = 85.9 and Fay-iddell s [3] stagnation heating q F- =550. W/cm (smaller than 67 W/cm in [33] and AUA s rediction [5] of 590 W/cm ). An examle of comuted flowfield is shown in Fig. 3b. oe (E-Fix) calculation diverged thus only the results of Van eer AUS+ and AUSPW+ are shown in Figs. 4 and 5 including the reference result from [5] where Gnoffo s version of oe flux was used which can control dissiation through multidimensional entroy fix. Surface ressure rofiles shown in Fig. 4 are almost symmetric and in good agreement with reference data. AUS+ yielded a very slight glitch at 5º (data at an angle stand for all the circumferential data in this test) but this is suressed by D term in AUSPW+. Calculated heating however are totally underestimated (Van eer) or strongly asymmetric (AUS+ and AUSPW+) as shown in Fig. 5 even with this hexahedral mesh. Gnoffo [5] recently develoed a multidimensional version of oe flux and the results shown therein were much better than the original oe though asymmetry was also seen in heating. AUSPW+ desite having a D dissiation term was not much better than AUS+ at reserving symmetry. In addition the heating contours of our results (Figs. 5b and 5c) showed different shaes from ef. [5] (or Fig. 5d) ossibly due to the use of different imlicit time integration methods and entire code structures. In this sense Gnoffo s latest udates [34] to his multi-dimensional flux would be romising thanks to multi-dimensional entroy-fix although it still seems to have left some exloration as a future challenge esecially in obtaining fully symmetric and accurate heating in the hyersonic shere roblem. Private communication with Peter Gnoffo and Bil Kleb NASA angley esearch Center ar. 009.

15 Figure 6 shows density residual histories of the resent comutations and AUS+ and AUSPW+ exhibited around three orders dro in the residual. From the engineering oint of view these solutions can be regarded as converged and thus we stoed our calculations (although even with this level of residual decrease it is seculated that shock anomalies grow later [9]). V. Final emarks We made a comarative study on the behaviors of flux functions with regard to two- and three-dimensional shock anomalies (instabilities and oscillations). The following features are noteworthy for hyersonic flow comutations in three dimensions (3D):. A simle exectation that three-dimensional shock anomalies always aear more likely than twodimensional (D) counterart turned out to be false. ather the develoment of shock anomalies is seen in every direction and quite comlicated in three dimensions. For instance in a two-dimensional setu in three dimensions (circular-cylinder in D) comutations demonstrated the following: a. A 3D case that exhibited a D carbuncle: The carbuncle develoed in the two-dimensional slice while the shock oscillation aeared in the third direction. These anomalies develoed from the very beginning of the comutation with the same growth rate. b. 3D cases that were stable in D: Deending on the grids or flux functions either the following two solutions were obtained: i) the totally symmetric solution or ii) the stable symmetric solution remained in the two-dimensional slice whereas the shock oscillation develoed in the third direction.. ultidimensional dissiations considered in AUSPW+ flux function worked to suress anomalous behaviors in limited cases but were not effective for genuinely two-dimensional or a genuinely threedimensional develoment of shock anomalies. 3. AUS-tye fluxes generally yielded satisfactory redictions of heating for a two-dimensional roblem in three dimensions for a cylinder but not for a genuinely three-dimensional roblem for a shere. It is demonstrated that multidimensional dissiation is effective but not erfectly. This is artly because such dissiation terms had been develoed under two-dimensional considerations and artly because those terms do not always work successfully even in two dimensions. Consequently a flux function showing good or fair robustness against the shock in two dimensions can either succeed or fail to reroduce accetable solutions in three dimensions. Thus when one attemts to test a flux function it is recommended to kee it in mind that investigations only in two

16 dimensions are not enough to accurately redict behaviors of the flux in three dimensions. Therefore although most of the existing Euler fluxes were designed based on one- or two-dimensions but readily extendable to threedimensions their erformances in 3D are too comlicated to be redicted from the D counterarts. Any flux function that eventually emerges as universal will have to meet (at the very least) all of the tests resented here. These are enough to show that a great variety of effects are ossible and that these are sensitive to many asects of the comutation. At the resent stage accetable heating rediction may be made by carefully considering those asects with a great effort or only by an exert user. Acknowledgments We are grateful to Peter Gnoffo NASA angley for giving us comutational grids used in the three-dimensional cases and for ermission to resent his numerical results as reference. We also thank Jeffery White and Bil Kleb NASA angley for roviding us with grids. The comutational code was originally develoed at Nagoya University Jaan while the first author was under suervision of Yoshiaki Nakamura. Hiroaki Nishikawa NIA (National Institute of Aerosace) gave us valuable comments. We thank their cooeration. eferences [] Peery K.. and Imlay S.T. Blunt-Body Flow Simulations AIAA Paer [] obinet J.-Ch. Gressier J. Casalis G. and oschetta J.-. Shock Wave Instability and the Carbuncle Phenomenon: Same Intrinsic Origin? J. Fluid ech. Vol. 47 (000) [3] Coulombel J.F. Benzoni-Gavage S. and Serre D Note on a Paer by obinet Gressier Casalis & oschetta J. Fluid ech. Vol. 469 (00) [4] oe P. Vorticity Caturing AIAA Paer [5] amalho.v.c. and Azevedo J..F. A Possible echanism for the Aearance of the Carbuncle Phenomenon in Aerodynamic Simulations AIAA Paer [6] Elling V. The Carbuncle Phenomenon is Incurable Acta athematica Scientia Vol. 9B No. 6 (009) [7] Pandolfi. and D Ambrosio D. Numerical Instabilities in Uwind ethods: Analysis and Cures for the Carbuncle Phenomenon Journal of Comutational Physics Vol. 66 No [8] Barth T. J. Some Notes on Shock-esolving Flux Functions Part : Stationary Characteristics NASA T [9] Kitamura K. oe P. and Ismail F. Evaluation of Euler Fluxes for Hyersonic Flow Comutations AIAA Journal

17 Vol [0] Kim K.H. Kim C. and ho O.H. Cures for the Shock Instability: Develoment of a Shock-Stable oe Scheme Journal of Comutational Physics Vol. 85 No doi:0.06/s00-999(0) [] Kim S.S. Kim C. ho O.H. Hong S.K. ethods for the Accurate Comutations of Hyersonic Flows I. AUSPW+ scheme Journal of Comutational Physics Vol [] Nishikawa H. and Kitamura K. Very Simle Carbuncle-Free Boundary-ayer esolving otated-hybrid iemann Solvers Journal of Comutational Physics Vol. 7 (008) [3] Shima E. and Kitamura K. Parameter-Free Simle ow-dissiation AUS-Family Scheme for All Seeds AIAA Journal Vol.49 No doi:0.54/ [4] oe P.. and Kitamura K. Artificial Surface Tension to Stabilize Catured Shockwaves AIAA Paer [5] Gnoffo P.A. ultidimensional Inviscid Flux econstruction for Simulation of Hyersonic Heating on Tetrahedral Grids AIAA Paer [6] oh C.Y. and Jorgenson P.C.E. ulti-dimensional Dissiation for Cure of Pathological Behaviors of Uwind Scheme Journal of Comutational Physics Vol. 8 (009) [7] Phongthanaanich S. and Dechaumhai P. Healing of Shock Instability for oe s Flux-Difference Slitting Scheme on Triangular eshes Int. J. Numer. eth. Fluids Vol. 59 (009) [8] Huang K. Wu H. Yu H. and Yan D. Cures for Numerical Shock Instability in HC Solver Int. J. Numer. eth. Fluids Vol. 65 (00) doi: 0.00/fld.7 [9] Gnoffo P.A. and White J.A. Comutational Aerothermodynamic Simulation Issues on Unstructured Grids AIAA Paer (004). [0] Candler G.V. avrilis D.J. Treviño. Current Status and Future Prosects for the Numerical Simulation of Hyersonic Flows AIAA Paer [] Kitamura K. Shima E. Nakamura Y. and oe P. Evaluation of Euler Fluxes for Hyersonic Heating Comutations AIAA Journal Vol.48 (00) doi:0.54/ [] Yoon S.H. Kim C. and Kim K.H. ulti-dimensional imiting Process for Three-Dimensional Flow Physics Analyses Journal of Comutational Physics Vol. 7 (008) [3] Balsara D.S. ultidimensional HE iemann Solver: Alication to Euler and agnetohydrodynamic Flows Journal of Comutational Physics Vol. 9 (00) [4] Van Albada G.D. Van eer B. and oberts Jr. W.W. A Comarative Study of Comutational ethods in Cosmic Gas Dynamics Astron. Astrohys. Vol. 08 (98)

18 [5] Van eer B. Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov s ethod Journal of Comutational Physics Vol. 3 (979) [6] oe P.. Aroximate iemann Solvers Parameter Vectors and Difference Schemes Journal of Comutational Physics Vol. 43 (98) [7] Harten A. High esolution Schemes for Hyerbolic Conservation aws Journal of Comutational Physics Vol. 49 (983) [8] Van eer B. Flux Vector Slitting for the Euler Equations ecture Notes in Phys. Vol. 70 (98) [9] iou.-s. A Sequel to AUS: AUS+ Journal of Comutational Physics Vol. 9 (996) [30] Hänel D Schwane. and Seider G. On the Accuracy of Uwind Schemes for the Solution of the Navier-Stokes Equations AIAA Paer (987). [3] en shov I.S. and Nakamura Y. Numerical Simulations and Exerimental Comarisons for High-Seed Nonequilibrium Air Flows Fluid Dynamics esearch Vol. 7 (000) [3] Fay J.A. and iddell F.. Theory of Stagnation Point Heat Transfer in Dissociated Air J. Aeronautical Sciences Vol. 5 (958) [33] Candler G. Barnhardt. Drayna T. Nomelis I. Peterson D. and Subbareddy P. Unstructured Grid Aroaches for Accurate Aeroheating Simulations AIAA Paer June 007. [34] Gnoffo P.A. Udates to ulti-dimensional Flux econstruction for Hyersonic Simulations on Tetrahedral Grids AIAA Paer

19 Figures =6 a) b) c) Figure. Tyical solutions for -/-D test a) Successful (hysically correct and stable) b) Unaccetable (shock oscillation ) c) Failure (shock instability called carbuncle ). Figure. Two-dimensional converged carbuncle solution. =6.0 i shock =+ j i a) b) Figure 3. -/-dimensional steady shock test (Test #A 50 5 cells) a) grid b) result of Van eer s FVS [9 ]. Solutions of this kind might be accetable for Euler simulations but inevitably contaminate surface heating in viscous comutations.

20 a) b) Figure 4. odified -/-dimensional test (Test #B cells) a) grid b) result of Van eer s FVS [].

21 Third Dimension (Direction) a) b) Figure 5. Three-dimensional steady lanar shock test (Test #C cells) a) grid b) oe (E-Fix) = stes (colors: ach number; gray: =.5 iso-surface). a) b) c) Figure 6. Three-dimensional steady lanar shock test (Test #C cells) Van eer a) = stes b) = stes c) = stes.

22 a) b) Figure 7. Three-dimensional steady lanar shock test (Test #C cells) AUS+ a) = stes b) = stes. regular bums a) b) c) Figure 8. Three-dimensional steady lanar shock test (Test #C cells) AUSPW+ a) = stes b) = stes c) = stes. a) b) c) Figure 9. Three-dimensional steady lanar shock test (Test #D cells) Van eer a) = stes b) = stes c) = stes (side view).

23 random bums a) b) c) d) Figure 0. Three-dimensional steady lanar shock test (Test #D cells) AUSPW+ a) = stes b) = stes (blow-u view) c) = stes and d) = stes.

24 Third Dimension (Direction) a) b) c) d) Figure. a) Grids and eference D Carbuncle esult [9] and b-g) Snashots of develoing 3D carbuncle for circular-cylinder the oe E-Fix first-order both in sace and time; freestream ach number =6.0); C contours (left; 0 < C < 50) and iso-ach-number surface (right; =.5): b) 500 stes c) 500 stes d) 000 stes e) 500 stes f) 5000 stes and g) stes continued.

25 e) f) D carbuncle and 3D oscillations g) Figure. a) Grids and eference D Carbuncle esult [9] and b-g) Snashots of develoing 3D carbuncle for circular-cylinder the oe E-Fix first-order both in sace and time; freestream ach number =6.0); C contours (left; 0 < C < 50) and iso-ach-number surface (right; =.5): b) 500 stes c) 500 stes d) 000 stes e) 500 stes f) 5000 stes and g) stes concluded.

26 -norm of density residual -norm of velocity asymmetry.e+00.e-0.e-04.e-06.e-08.e-0.e-.e-4 oe (E-Fix) Vaneer AUS+.E+0.E+00.E-0.E-04.E-06.E-08.E-0.E-.E-4 dv w a).e time stes b).e time stes Figure. esidual and asymmetry histories for 3D circular-cylinder ( =6.0) a) Density residual and b) Velocity asymmetry in v and w comonents. a) b) c) Figure 3. a) Grid and eference D Stable esult (AUS+) [9] and 3D results by C contours (0 < C < 47) for circular-cylinder at stes first-order both in sace and time =6.0: b) AUS+ and c) Van eer s FVS.

27 a) b) c) d) Figure 4. Pressure contours (0 < P/P < 386) for D circular-cylinder (second-order in sace; freestream ach number =7). [Original grid =0.0; cells]: a) oe (E-Fix) b) Van eer c) AUS+ d) AUSPW+. a) b) c) d) Figure 5. Pressure contours (0 < P/P < 386) for D circular-cylinder (second-order in sace; freestream ach number =7). [odified (half-cell dilated) grid =0.5; cells]: a) oe (E-Fix) b) Van eer c) AUS+ d) AUSPW+.

28 P/P oe (E-Fix) Vaneer Hanel AUS+ AUSPW+ ef. (AUA) P/P oe (E-Fix) Vaneer Hanel AUS+ AUSPW+ ef. (AUA) [deg.] [deg.] a) a) q/q F oe (E-Fix) Vaneer Hanel AUS+ AUSPW+ ef. (AUA) q/q F oe (E-Fix) Vaneer Hanel AUS+ AUSPW+ ef. (AUA) [deg.] [deg.] b) b) Figure 6. Surface ressure and heating rofiles for D cylinder (second-order in sace; freestream ach number =) [Original grid =0.0]: a) Pressure and b) Heating. Figure 7. Surface ressure and heating rofiles for D cylinder (second-order in sace; freestream ach number =) [odified (half-cell dilated) grid =0.5]: a) Pressure and b) Heating.

29 only third directional oscillations a) b) c) d) Figure 8. Pressure (to; 0 < P/P < 390) and ach number (bottom; 0 < < 7) contours for 3D circularcylinder (second-order in sace; freestream ach number =7): a) oe (E-Fix) b) Van eer c) AUS+ and d) AUSPW+.

30 P/P oe (E-Fix) 3D P/P Vaneer 3D 0.4 oe (E-Fix) D 0.4 Vaneer D 0. ef. (AUA) D 0. ef. (AUA) D a) [deg.] a) [deg.].0. oe (E-Fix) 3D.5 oe (E-Fix) D.0 ef. (AUA) D 0.8 Vaneer 3D q/q F.0 q/q F 0.6 Vaneer D 0.4 ef. (AUA) D [deg.] [deg.] b) b) Figure 9. Surface ressure and heating rofiles for 3D cylinder (second-order in sace; freestream ach number =) [Original grid =0.0 oe (E-Fix)]: a) Pressure and b) Heating. Figure 0. Surface ressure and heating rofiles for 3D cylinder (second-order in sace; freestream ach number =) [Original grid =0.0 Van eer]: a) Pressure and b) Heating.

31 P/P AUS+ 3D P/P AUSPW+ 3D 0.4 AUS+ D 0.4 AUSPW+ D 0. ef. (AUA) D 0. ef. (AUA) D [deg.] [deg.] a) a) q/q F 0.6 AUS+ 3D q/q F 0.6 AUSPW+ 3D 0.4 AUS+ D 0.4 AUSPW+ D 0. ef. (AUA) D 0. ef. (AUA) D [deg.] [deg.] b) b) Figure. Surface ressure and heating rofiles for 3D cylinder (second-order in sace; freestream ach number =) [Original grid =0.0 AUS+]: a) Pressure and b) Heating. Figure. Surface ressure and heating rofiles for 3D cylinder (second-order in sace; freestream ach number =) [Original grid =0.0 AUSPW+]: a) Pressure and b) Heating.

32 a) b)..0 AUSPW+ 0.8 ef. (AUA-B axisymmetric) P/P [deg.] d) c) Figure 4. Surface ressure contours (to; 0 < P/P < 90) and rofiles (bottom) (3D shere): a) Van eer b) AUS+ c) AUSPW+ and d) ef. [5] courtesy of Gnoffo.

33

34 Vaneer 0.8 ef. (AUA-B axisymmetric) AUS+ q/q F q/q F ef. (AUA-B axisymmetric) [deg.] [deg.] b) a).4. AUSPW+ q/q F.0 ef. (AUA-B axisymmetric) [deg.] d) c) Figure 5. Surface heating contours (to; 0 < q < 600 [W/cm]) and rofiles (bottom) (3D shere): a) Van eer b) AUS+ c) AUSPW+ and d) ef. [5] courtesy of Gnoffo.

35 -norm of density residuals a) Grid b) Pressure contours (AUS+) Figure 3. Comutational grid and tyical solution for = Candler s challenge roblem [34]..E-0.E-0.E-03 Van eer AUS+ AUSPW+.E-04.E-05.E time stes Figure 6. esidual histories for shere ( =).

36 #4. Viscous Shere = #3. Viscous Cylinder =7 #. Cylinder =6 #. Normal Shock =6 Tables Table. Shock anomalies Anomalies Oscillations Instabilities D Y (only in time) N (Not observed yet) D Y (both in time and sace) Y (Carbuncle) Table. Classification of Euler Fluxes Based on Three Proerties for Hyersonic Heating [] Flux Functions Grou Grou Grou 3 oe (E-Fix) Van eer AUS+ AUSPW+ I. Shock Stability/obustness Poor Good Fair II. H-reserving N N Y III. B- esolution Y N Y Table 3: Test cases and results S (Successful: stable and symmetric) U (Unaccetable: oscillatory asymmetric or oor rediction of heating) and F (Failure: carbuncle or diverged). Cases Euler Fluxes D 3D A=0.0 =0.5 B =0.0 C =0.0 =0.5 #D =0.0 oe (E-Fix) F F - F F - (Carbuncle) (Carbuncle) (Carbuncle) (Carbuncle) Van eer S S F S S S (Carbuncle) AUS+ S U (Not - S U (Not - converged) converged) AUSPW+ S S F(Carbunc S S F le) (Carbuncle) =0.0 =0.5 =0.0 =0.5 oe (E-Fix) F (Carbuncle) F (Carbuncle) F (Carbuncle) F (Carbuncle) Van eer S S S - AUS+ S S S - AUSPW+ S S S - =0.0 =0.5 =0.0 =0.5 oe (E-Fix) S U (Oscillatory) U (Oscillatory) F (Diverged) Van eer U (Severely low U (Severely low U (Severely low U (Severely low heating) heating) heating) heating) AUS+ U (Not converged) U (Not converged) U (Not converged) U (Not converged) AUSPW+ S S S S oe (E-Fix) F (Diverged) Van eer U (Severely low heating) AUS+ U (Asym.) AUSPW+ U (Asym.)