james n. moss
TRANSCRIPT
European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2000
Barcelona, 11-14 September 2000c ECCOMAS
DSMC SIMULATION OF SEPARATED FLOWS ABOUT
FLARED BODIES AT HYPERSONIC CONDITIONS
James N. Moss?
?Aerothermodynamics Branch, NASA Langley Research Center, Hampton, VA23681-2199, USA, email: [email protected], web page:
http://abweb.larc.nasa.gov:8080//
Key words: hypersonic, low density, DSMC computations, shock/shock interactions,shock/boundary layer interactions, comparisons with measurements
Abstract. This paper describes the results of a numerical study of interacting hypersonic ows at conditions produced in ground-based test facilities. The computations are madewith the direct simulation Monte Carlo (DSMC) method of Bird. The focus is on Mach10 ows about ared axisymmetric con�gurations, both hollow cylinder ares and doublecones. The ow conditions are those for which experiments have or will be performed in theONERA R5Ch low-density wind tunnel and the Calspan-University of Bu�alo ResearchCenter (CUBRC) Large Energy National Shock (LENS) tunnel. The range of ow con-ditions, model con�gurations, and model sizes provides a signi�cant range of shock/shockand shock/boundary layer interactions at low Reynolds number conditions. Results pre-sented will highlight the sensitivity of the calculations to grid resolution, contrast the dif-ferences in ow structure for hypersonic cold ows and those of more energetic but stilllow enthalpy ows, and compare the present results with experimental measurements forsurface heating, pressure, and extent of separation.
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1 INTRODUCTION
Hypersonic separated ows produce shock/shock and shock/boundary layer interac-tions that create augmented aerothermal loads and reduced surface control e�ectiveness,issues critical to hypersonic vehicle design. To enhance the understanding of such ows,experimental and computational studies have been actively promoted by the NATO Re-search Technology Organization (RTO, formerly AGARD) for basic axisymmetric con-�gurations and for a range of Reynolds numbers. The current study focuses on the lowReynolds number experimental conditions where separation and reattachment occur un-der laminar conditions and where the ow is assumed to be steady. Calculations aremade by using the direct simulation Monte Carlo (DSMC) method of Bird [1] for bothhollow cylinder- are and double-cone models. The ow conditions simulated are those forwhich experiments have been or will be conducted in two facilities: the ONERA R5ChMach 10 low-density wind tunnel and the Calspan-University of Bu�alo Research Center(CUBRC) Large Energy National Shock (LENS) tunnel.
The hollow cylinder- are model has a sharp leading edge, a cylinder 101.7 mm long,and a 30� are. The author has presented extensive calculations (refs. [2] - [7]) forthe cylinder- are model at the ONERA test conditions highlighting the noncontinuumand continuum aspects for the ow, the sensitivity to numerical simulation parameters,the agreement with measurements made in the ONERA wind tunnel, di�erences in two-dimensional (2D) and axisymmetric results, and comparisons with boundary layer (onlycylinder or plate results) and Navier-Stokes solutions. These results are reviewed alongwith the presentation of new results that demonstrate the sensitivity of the calculationsfor the cylinder portion of the model to numerical parameters. Also, results of newcalculations are included for more energetic ows, as produced in the LENS tunnel for anominal test condition (Mach 9.6 nitrogen). For the LENS test conditions, calculationsare presented for two models|the ONERA model con�guration and a model with a muchlonger are.
Calculations for the double-cone models [8] are for the same ow conditions as the hol-low cylinder- are study. For the double-cone con�gurations investigated, the shock/shockinteractions are stronger than those for the hollow cylinder- are models. The �rst conehalf angle is 25�, while the second cone half angle is either 55� or 65�. These doublecone geometries produce strong shock interactions because the attached shock from the�rst cone interacts with the detached bow shock from the second cone. Also, the outershocks are modi�ed by the separation and reattachment shocks where the extent of owseparation is signi�cant for the combinations of model size and ow conditions examined.Results are presented that demonstrate the sensitivity of the surface results to numericalparameters, Reynolds number, and ow conditions.
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2 DSMC Code
The DSMC code used in the current study is the general 2D/axisymmetric code ofBird [1],[9]. The molecular collisions are simulated with the variable hard sphere (VHS)molecular model. Energy exchange between kinetic and internal modes is controlled bythe Larsen-Borgnakke statistical model [10]. For the present study, the simulations areperformed by using nonreacting gas models while considering energy exchange betweentranslational, rotational, and vibrational modes. A constant rotational relaxation collisionnumber of 5 was used for the calculations. The vibrational collision number was 50. Thereference conditions for the VHS model were as follows: reference temperature = 300 K,temperature exponent of the viscosity coe�cient = 0.75, and reference diameters for O2
and N2 were 3.96 x 10�10 m and 4.07 x 10�10 m, respectively. The model surface isassumed to have a speci�ed constant temperature. Full thermal accommodation anddi�use re ection are assumed for the gas-surface interactions.
Common to the DSMC simulations presented is the treatment of the computationaldomain griding, which consisted of an arbitrary number of regions. Each region is subdi-vided into cells, and the cells in selected regions are subdivided into subcells to enhancethe spatial resolution used to select collision partners. In general, the cell dimensionswithin a region were nonuniform in both directions, with geometric stretching exceedingan order of magnitude in some regions. Also, the macroscopic quantities are time-averagedresults extracted from the individual cells. Since the computational regions were not runwith necessarily the same time step, it was essential that steady state conditions be es-tablished before generating the �nal time-averaged results. Steady state was assumed tooccur when the total molecules used in the simulation, average molecules in each region,and surface quantities (locations and size of the separation region, heating, etc.) becameessentially constant when sampled sequentially over signi�cant time intervals.
3 CALCULATIONS FOR HOLLOW CYLINDER FLARE
The ONERA hollow cylinder- are test case considered was formulated initially as oneof the test problems concerning shock wave/boundary layer interactions in an AGARDWorking Group 18 activity [11] for the validation of Navier-Stokes solvers for cold high-speed ows where the interactions produce large separated regions under laminar con-ditions. The initial test case generated considerable interest for code validation, as isevident by the AGARD activity [11] and several independent workshops. Interest in thisproblem continues with the expansion of test cases to include additional ow conditionsand model sizes [12],[13] in the current Research Technology Organization (RTO, formallyAGARD), working group 10 activities.
A motivation for investigating these test cases with DSMC has been to identify thelevel of grid resolution and related computational parameters that one must use to achieveaccurate results for problems with complex interactions (where the grid resolution is im-portant in directions other than the one normal to the surface). Furthermore, DSMC
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codes provide a simulation capability that is valid across the entire ow spectrum of freemolecular through continuum, a desirable feature for the current problems. However,there are practical limitations when using the DSMC approach due to excessive computa-tional requirements as one moves well into the continuum regime for multiple-dimensionalproblems.
The current study includes numerical simulations for two sets of ow conditions andtwo hollow cylinder- are models. Table 1 lists the free-stream and surface conditions forthe experiments that have been conducted in the ONERA R5Ch low-density wind tunnel.Also included is a set of nominal ow conditions for a test scheduled to be performed inthe LENS tunnel.
Details of the model con�guration used in the ONERA tests are presented in Fig. 1.The hollow cylinder has a sharp leading edge with a bevel angle of 15� and is alignedparallel to the oncoming ow. The compression are is inclined 30� to the cylinder andis terminated by a hollow cylindrical section. Additional information concerning modelconstruction, materials, and instrumentation is given in refs. [14] and [6].
For the CUBRC tests, two hollow cylinder- are con�gurations will be used as describedin [12] and [13]|one that mimics the ONERA model (current calculations use the dimen-sions shown in Fig. 1 for describing the model outer surface) and a model with a muchlonger are. For the long- are model, the horizontal length of the are is 118.28 mmrather than the 43.3 mm shown in Fig. 1, and the model terminates at the end of the are.
Table 1: Free-stream and surface conditions
V1, �1, n1, T1, p1, TW ,Facility m/s kg/m3 m�3 K N/m2 Gas M1 KONERA R5Ch 1418.7 4.303 � 10�4 0.895 � 1022 51.0 6.30 Air 9.91 293.0
CUBRC LENS 2718.6 6.808 � 10�4 1.463 � 1022 194.1 39.21 N2 9.56 297.8
3.1 Previous Results and Findings for the ONERA R5Ch Tests
Figures 2 through 8 present results of the calculations �rst reported in refs. [4] and [5]that describe the ow-�eld features and surface results for the ONERA hollow cylinder- are test case. The experimental value for free-stream Reynolds number is 18 916, wherethe viscosity (3.29 x 10�6 Pa � s) is given by the Sutherland expression and the charac-teristic dimension is the cylinder length L. Also presented are comparisons of the surfaceresults for heating, pressure, and the extent of separation with the experimental measure-ments described in refs. [14] , [6], and [7]. The current results are those obtained withthe �nest grid resulting from the grid resolution study described in ref. [4]. The previouscalculations show that the extent of separation is quite sensitive to the grid|a muchsmaller separation region is obtained with a coarse grid. Data included in Fig. 2 provide
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information concerning the grid and simulation parameters. A four-region computationaldomain was used where each cell was subdivided into four subcells (2 x 2). The time stepin each of the four regions had values of 75, 75, 28, and 15 ns, respectively.
General ow features for this test case are evident in Figs. 2 and 3. Figure 2 showsselected Mach contours and streamlines while Fig. 3 presents the normalized densitycontours. Evident is a large separation region characterized by a single vortex embeddedin the subsonic ow region. Calculated locations for separation and reattachment (denotedby S and R, respectively) are 76.76 mm and 134.4 mm downstream of the cylinder leadingedge. The shock/shock interaction occurs near the end of the are where the shock layerthickness is at a minimum and the density is at a maximum, equal to 14.4 times thefree-stream value.
30°20°
22.5
101.7 43.3 25
32.5
57.5
15°
Dimensions in mm
������
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S
R
M = 9.8
12
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4
Region Cells Subcells/Cell1 75 x 100 2 x 22 100 x 225 2 x 23 120 x 375 2 x 24 20 x 180 2 x 2
M = 1
Mach 9.91 air, Re∞,L =18,916
Fine-grid results4-region domain78,100 cells312,400 subcells1,900,574 molecules∆t1 = 75 nst1,avg = 3.67 to 5.25 ms
L = 101.7 mmxS = 76.76 mmxR = 134.4 mm∆x/L = 0.567
Figure 1: Hollow cylinder- are model ( x mea-sured from leading edge and L = 101.7 mm).
Figure 2: Flow structure and simulation pa-rameters for ONERA hollow cylinder are.
0.00 0.05 0.10 0.15x, m
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ρ = 4.303 x 10-4 kg/m3
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0.76 1.34
ONERA R5ChSeparation region
DSMC results
End of flare
Figure 3: Density contours for ONERA hollowcylinder are.
Figure 4: Skin friction coe�cient and extent ofseparation (oil- ow data from experiments).
Figures 4 through 6 present the calculated surface results for skin friction coe�cient,heating rate, and pressure coe�cient as a function of the distance from the cylinder
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James N. Moss
leading edge (normalized by the cylinder length L). Maximum values for friction, heatingrate, and pressure occur on the are at a location downstream of reattachment|veryclose to the end of the are located at x/L = 1.426. Included in these �gures are theresults of the experimental measurements [6], [7] for the extent of separation as inferredfrom oil ow measurements, heating rates extracted by using a thin-�lm technique, andsurface pressure inferred from variable reluctance di�erential transducers connected tomodel pressure taps by tubes.
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75x/L
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q,kW
/m2
DSMC
ONERA R5ChRun numbers
1022102610281029
End of flare
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9829839849909929951030116411661167DSMC
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ONERA R5ChRun numbers
End of flareCp = 2(pW - p∞)/ρ∞V∞2
Figure 5: Heat-transfer rate distributions. Figure 6: Pressure coe�cient distributions.
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**************************************Experimental results adjusted by
a factor of 1.4**************************************
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9829839849909929951030116411661167DSMC
#
+*
**************************************Experimental data adjusted by a
factor of 1.4**************************************
ONERA R5ChRun numbers
Figure 7: Pressure coe�cients|experimentaldata adjusted.
Figure 8: Cylinder pressure coe�cients|experimental data adjusted.
Agreement between the calculated and the measured results are very good for theextent of separation and the heating rate distribution; however, there are noticeable dif-ferences for the pressure distribution. The separation location is the same for both setsof results. The calculated reattachment location occurs, however, somewhat forward of
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the experimental value, and the extent of the calculated separation (�x/L) is 98% of themeasurement. The heating-rate distributions are characteristic of those for laminar owsin that the heating shows an initial decrease at the location of separation (�x/L = 0.76),a cusp-like behavior at the juncture, and a rapid increase along the are.
Among the surface quantities, the agreement between the current calculations andmeasurements is the poorest for pressure. This discrepancy is particularly puzzling sincethe agreement for both heat transfer and the locations for separation and reattachmentare very good. The trends of the two data sets are qualitatively consistent; however, thecomputational results are consistently higher than the measured values. As �rst pointedout in Ref. [4], the 42% discrepancy near the peak pressure location on the are is veryobvious; however, di�erences of this magnitude are also present along the hollow cylinder.In fact, this di�erence is a constant. If the experimental pressure values are multipliedby a factor of 1.4, agreement between the two data sets becomes very good, as shown inFigs. 7 and 8, where Fig. 8 presents an enlarged view focusing primarily on the cylinder.Additional results are presented in section 3.2 that address the lack of agreement alongthe cylinder between calculation and measurement.
Flow-�eld density measurements have also been performed at ONERA for the hollowcylinder- are model by detecting X-ray emissions from the gas produced by electron beamimpact. The experimental results have been compared [7] with numerical results obtainedby using the current DSMC results and two Navier Stokes codes. The agreement betweenmeasurements and calculations is somewhat mixed since neither DSMC nor Navier Stokesresults provided consistent agreement with the measurements made at three locations(x/L = 0.3, 0.6 and 0.76) along the cylinder. Details concerning these measurements andcomparisons are given in Ref. [7].
3.2 E�ect of Computational Parameters on Hollow Cylinder Results
This section presents the results of calculations for only the hollow cylinder portion ofthe ONERA model to address the discrepancy observed between measured and calculatedsurface pressures. With con�dence in the ability to computationally resolve and accuratelypredict the ow conditions along the cylinder, additional calculations were made thatfocus on the sensitivity of the calculated results to numerical parameters. The focus ison the cylinder terminated at x = 70 mm, the interface location between regions 1 and2 (Fig. 2). The purpose of this portion of the study is to examine the sensitivity ofthe surface and ow-�eld results to variations in the numerical parameters used in theresults discussed earlier; in particular, to determine if the calculated surface pressure isin uenced by additional re�nements and leading edge treatments (amount of ow domainincluded in the computational domain and grid resolution). The particular parameters forwhich variations were made are the magnitude of the time step, grid resolution, and thetreatment given to the model leading edge. As shown in Figs. 9 though 12, the results forthese cases are compared to each other and to the hollow cylinder- are results previouslypresented.
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Hollow cylinder-flare (H.C.F.) results and five (1 through 5)solutions for hollow cylinder with changes in time step,grid resolution, and grid domain near the leading edge.
p∞ = 6.30 N/m2
TW = 293 KCases
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Hollow cylinder-flare (H.C.F.) results and five(1 through 5) solutions for hollow cylinderwith changes in time step, grid resolution,and grid domain near the leading edge.
TW = 293 K
Cases
Figure 9: E�ect of simulation parameters oncylinder pressure distributions.
Figure 10: E�ect of simulation parameters oncylinder heating-rate distributions.
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Cases
Hollow cylinder-flare (H.C.F.) resultsand three cylinder cases that accountfor refinements in grid resolution,time step, and leading edgetreatment. Cylinder ends at x = 70 mm.Surface is located at y = 32.5 mm.
x = 30.5 mmx/L = 0.3
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125H.C.F.
Cases
Hollow cylinder-flare (H.C.F.) resultsand three cylinder cases that accountfor refinements in grid resolution,time step, and leading edgetreatment. Cylinder ends at x = 70 mm.Surface is located at y = 32.5 mm.
x = 61 mmx/L = 0.6
Figure 11: E�ect of simulation parameters oncalculated density pro�les (x/L = 0.3).
Figure 12: E�ect of simulation parameters oncalculated density pro�les (x/L = 0.6).
Surface results for pressure and heating rate are given in Figs. 9 and 10, respectively.The in uence of the are on the hollow cylinder- are (denoted as H.C.F in Figs. 9-11)results extends slightly upstream of the x = 70 mm location, clearly evident in the surfacepressure distribution (Fig. 9) but not evident in the heating-rate distribution (Fig. 10).Also, the out ow boundary condition imposed (free stream at the end of the cylinder)has an in uence on the surface pressure results that extends about 6 mm upstream of thecylinder termination. The parametric variations for each of the �ve cylinder calculationsare as follows: Case 1 was a one-region computational domain identical to that used in theH.S.F. simulation (see Fig. 2); for Case 2, the number of cells and subcells were increasedby a factor of three (140 x 150 cells), and the time step was 20% of that for Case 1 (15 ns);Case 3 used a time step of 25 ns and the same cell resolution as H.C.F. and Case 1; Case 4
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used a time step of 25 ns and the same grid resolution as the H.C.F. but had two extraregions, with additional treatment in front of and downstream of the leading edge (aregion upstream of the leading edge [-3 mm] and a region that extended a short distance[3 mm] downstream of the leading edge with �ve times the �x resolution of the H.S.Fcase). Case 5 di�ered from Case 4 in that it included two additional regions to accountfor the in uence of the beveled leading edge underside. Common to all the solutions werethe four subcells/cell and a scaling of real to simulated molecules such that there were,on average, approximately 25 simulated molecules per cell. When the surface results forpressure and heating rate distributions are compared with the H.S.F. results, there is noe�ect other than the expected results very near the leading edge|as the cell dimension inthe x-direction (�x) decreases near the leading edge, one gets an improved de�nition ofthe surface quantities where a local maximum occurs and then decreases in value as theleading edge is approached (quantities are decreasing toward their free molecular valuesbut will not achieve the free molecular values because of upstream in uence). As for theimpact of these additional re�nements on the downstream ow-�eld quantities, no impactis evident on the density pro�les at x/L locations of 0.3 and 0.6, as shown in Figs. 11 and12, respectively.
Based on the �ndings of this section, it is believed that the DSMC results presentedearlier for surface pressure are correct for the cylinder and should be reasonably accuratefor the are, based on a constant discrepancy of 40% with measurements for both cylinderand are. Also, the current DSMC results are in good agreement with those obtained byMarkelov et al. [16] for the ONERA test case, in which a di�erent DSMC code was used.
3.3 Computational Results for LENS Flow Conditions
This section presents results of DSMC calculations for ow about two hollow cylinder- are models at a nominal LENS ow condition (Table 1), where the ow is Mach 9.57nitrogen at a free-stream Reynolds number of 14 920 (characteristic length is the cylinderlength, L = 101.7 mm, and the viscosity is given by the VHS [1] model). The ow ismore energetic than the R5Ch conditions; however, the ow enthalpy is still quite lowand chemical reactions are neglected for this test case condition. As previously discussed,the short are model was assumed to have the same outer surface dimensions as theONERA model (Fig. 1) while the long- are model has simply an extended are with themodel terminated at the end of the are. The long- are model was included [12] in theexperimental program to allow for a complete pressure recovery on the are, thus makinga more straightforward comparison with theoretical models. The current results providean indication of the sensitivity of the DSMC calculations to grid resolution, show thatthe extent of separation is much smaller than that for the R5Ch ow conditions, showthat the results for the long and short are are essentially identical within the domainscommon to the two models, and provide information concerning the general features ofthe surface results and ow-�eld structure.
Figures 13 and 14 provide information describing the computational domain, simula-
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James N. Moss
tion parameters, ow structure (selected Mach contours and streamlines), and calculatedlocations for separation and reattachment for the short and long are CUBRC models,respectively. (The symbol F in Figs. 13 and 14 denotes the ratio of real to simulatedmolecules.) These results are for the �nest grid calculations resulting from several com-putations where grid re�nement and sensitivity studies were performed. Figure 15 is anexample of the results for the long are model showing the sensitivity of heating rate todi�erent combinations of regions, cells, and subcells|a factor of 27 in subcell resolution.The e�ect of grid resolution on heating shows the expected trend of decreased heatingwith improved grid resolution outside the surface areas in uenced by ow separation.Also, the size of the separation zone increases with improved grid resolution, as indicatedby the tabulated results for �x/L included in Fig. 15. The peak heating downstream ofreattachment is slightly higher for the �ner grid results.
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Fine-grid results4-region domain85,600 cells964,900 subcells1,281,012 molecules∆t1 = 20 nst1,avg = = 2.72 to 3.76 ms
F1 = 2.3 x 1013
Region F/F1 ∆t/∆t1 Cells Subcells1 1.00 1.00 70 x 100 2 x 22 0.35 1.00 100 x 225 2 x 33 0.25 0.25 140 x 375 5 x 34 1.50 0.20 20 x 180 2 x 2
1 23
4
xS = 91.1 mmxR = 118.1 mm
M∞ = 9.57 nitrogenRe∞,L = 14,920
L = 101.7 mm
RS
M = 9
M = 1
∆x/L = 0.265
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Region F/F1 ∆t/∆t1 Cells Subcells1 1.00 1.00 70 x 100 2 x 22 0.35 1.00 100 x 225 2 x 33 0.25 0.25 150 x 375 5 x 34 0.50 0.25 150 x 375 3 x 2
Fine-grid results4-Region domain142,000 cells1,344,250 subcells2,231,192 molecules∆t1 = 20 nst1,avg = 2.60 to 2.76 ms
F1 = 2.3 x 1013
M∞ = 9.57 nitrogenRe∞,L = 14,920
L = 101.7 mm
xS = 91.1 mmxR = 117.9 mm∆x/L = 0.263
RS
M = 9
M = 1
Figure 13: Flow structure and solution parametersfor CUBRC model with short are.
Figure 14: Flow structure and solution parametersfor CUBRC model with long are.
The results presented in Fig. 15 do not demonstrate grid convergence; however, resultsfor the short are model, where the grid was identical to that for the long are modelfor regions 1 through 3, showed that the surface results were in close agreement for twosolutions, where one solution had the cell/subcell arrangement given in Fig.13 (same asfor the long are solution for regions 1 through 3) and one with the same number of cells(85 600) but only about half the number of subcells (531 200).
Comparisons of surface results for the short and long are models are presented forheating coe�cient, pressure coe�cient, and skin friction coe�cient in Figs 16, 17, and18, respectively. The results show that the surface quantities are essentially identical forsurfaces common to the short and long are models. For this particular LENS nominaltest condition, the extent of separation is only 47% of that calculated and measured forthe ONERA R5Ch test condition. With the smaller extent of separation, the calculations
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suggest that the short are is su�ciently long to negate any in uence of the expansionat the end of the are on the location of reattachment. However, the end of are ex-pansion produces a thinning of the boundary layer and signi�cant changes in the surfacequantities|decreasing pressure (Fig. 17) and increasing heating (Fig. 16) and friction(Fig. 18). The same trends are evident for the short- are model where the ow expandson to the cylindrical extension at x = 0.145, as is clearly evident in the heating and frictionresults (also examine Fig. 19).
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5 49,625 49,625 0.1995 49,625 147,500 0.2224 142,000 1,344,250 0.263
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Flare, Grid for 4-region domainsCells Subcells
Short 85,600 964,900Long 142,000 1,344,250
CH = q /( 0.5 ρ∞ V∞3)
0.5 ρ∞ V∞3 = 6,840 kW/m2
Figure 15: E�ect of grid on heating rate|long are at a LENS ow condition.
Figure 16: Heat-transfer coe�cient results for anominal LENS test condition.
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Cp= (pW - p∞) / (0.5 ρ∞ V2∞)
0.5 ρ∞ V∞2 = 2,516 N/m2
p∞ = 39.2 N/m2
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Flare, Grid for 4-region domainsCells Subcells
Short 85,600 964,900Long 142,000 1,344,250
Cf = τ / (0.5 ρ∞ V2∞)
0.5 ρ∞ V∞2 = 2,516 N/m2
Figure 17: Pressure coe�cient results for a nomi-nal LENS ow condition.
Figure 18: Skin friction coe�cient results for anominal LENS ow condition.
Composite plots for heating rate, pressure, and skin friction are given in Figs. 19 and20 for the short and long are models, respectively. These data sets provide information
11
James N. Moss
on the correlation of the three surface quantities and explain how they are in uenced byseparation. The general qualitative features are the same as previously discussed for theR5Ch test conditions.
0.0 0.5 1.0 1.5
x/L
0
50
100
150
200
250
300
350
400
q,kW
/m2
0
500
1000
1500
2000
2500
pan
dτ,
N/m
2
pτq
(x/L)S = 0.896(x/L)R = 1.161
S
R
∆x/L = 0.265
0.0 0.5 1.0 1.5 2.0
x/L
0
50
100
150
200
250
300
350
400
q,kW
/m2
0
500
1000
1500
2000
2500
pan
dτ,
N/m
2
pτq
(x/L)S = 0.896(x/L)R = 1.159
S
R
∆x/L = 0.263
Figure 19: DSMC surface results for a nominalLENS condition (short are).
Figure 20: DSMC surface results for a nominalLENS condition (long are).
Figures 21 through 24 provide information describing the ow structure for the short are. Contours for nondimensional density, overall kinetic temperature, and scalar pres-sure (nkT, where T is the overall kinetic temperature) are included. The are-inducedadverse pressure gradient is evident in Fig. 23 where the isopressure lines coalesce into aseparation shock that compresses the ow to a maximum density of 29.6 times the free-stream value. An enlarged view of the scalar pressure contours in the reattachment regionand beyond are presented in Fig. 24. Along the surface and downstream of reattachmentare the locations for maximum density and scalar pressure, with magnitudes equal to29.6 and 57.2 times their respective free-stream values. When the current calculated re-sults for the LENS condition are compared with the calculated results for the R5Ch owconditions (see ref. [4]), the maximum density is 3.25 times greater and occurs at thesurface rather than in the shock layer as it does for the R5Ch case (Fig. 3). Also, thetemperatures are much higher, the maximum temperature being 2.8 times greater. Forthe maximum scalar pressure, the calculated LENS value is 6.2 times that calculated forthe R5Ch condition.
Figures 25 through 28 present calculated surface normal pro�les for the CUBRC modelwith a short are. Included are six body station pro�les along the cylinder where the sur-face is located at y = 32.5 mm. Data for four variables|density, scalar pressure, overallkinetic temperature, and tangential velocity|are presented. These pro�les help to iden-tify some of the thermal nonequilibrium aspects of the ow with evidence of temperatureand velocity jump at the surface, where the magnitude of the jump increases as the leadingedge is approached. Also note that for the nonequilibrium situations, the scalar pressurewill be di�erent from the normal force per unit area on an element of solid surface; that is,
12
James N. Moss
the surface normal force and the gas scalar pressure adjacent to the surface element willbe di�erent, as they are for the present case, particularly near the model leading edge.
0.00 0.05 0.10 0.15x, m
0.00
0.02
0.04
0.06
0.08
0.10
y,m
1.021.5
2 0.40.6
4
8
1.5
2
SR
(ρ/ρ∞)Max = 29.6
ρ∞ = 6.808 x 10-4 kg/m3
Max
0.00 0.05 0.10 0.15x, m
0.00
0.02
0.04
0.06
0.08
0.10
y,m
200
500
1000
SR
1500 K
TMax = 1743 K
T∞ = 194.1 K
1000 K
Figure 21: Density contours for a nominal LENScondition (short are).
Figure 22: Overall kinetic temperature contoursfor a LENS condition (short are).
0.00 0.05 0.10 0.15x, m
0.00
0.02
0.04
0.06
0.08
0.10
y,m
22 1.02
310
3 2S
R
[(nKT)/(nKT)∞]Max = 57.2
p∞ = 39.21 N/m2
Max
10 303
0.12 0.13 0.14 0.15x, m
0.045
0.050
0.055
0.060
y,m 2
20
50
5
55
1.02
3040
50
4020 10
540
2
10
R
[(nKT)/(nKT)∞]Max = 57.2
p∞ = 39.21 N/m2
Max
Figure 23: Scalar pressure contours for a nominalLENS condition (short are).
Figure 24: Scalar pressure contours along are fora nominal LENS condition (short are).
ρ/ρ∞
y,m
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.032
0.036
0.040
0.044
0.048
0.052 x/L = 1.0
0.76
0.6
0.3
0.2
0.1
ρ∞ = 6.808 kg/m3
L = 101.7 mmTW = 297.8 K
Surface is at y = 32.5 mm
(nkT)/(nkT)∞
y,m
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
0.032
0.036
0.040
0.044
0.048
0.052 x/L = 1.0
0.76
0.6
0.3
0.2
0.1
p∞ = 39.21 N/m2
L = 101.7 mmTW = 297.8 K
Surface is at y = 32.5 mm
Figure 25: Density pro�les for a nominal LENScondition (short are).
Figure 26: Scalar pressure pro�les for a nominalLENS condition (short are).
13
James N. Moss
T, K
y,m
0 500 1000 1500
0.032
0.036
0.040
0.044
0.048
0.052
x/L = 1.0
0.760.60.30.2
0.1
T∞ =191.4 KL = 101.7 mm
Surface is at y = 32.5 mm
TW = 297.8 K
u, m/sy,
m
-500 0 500 1000 1500 2000 2500 3000
0.032
0.036
0.040
0.044
0.048
0.052
x/L = 1.00.76
0.6
0.3
0.2
0.1
V∞ = 2718.6 m/sL = 101.7 mm
Surface is at y = 32.5 mm
Figure 27: Overall kinetic temperature pro�les fora nominal LENS condition (short are).
Figure 28: Tangential velocity pro�les for a nom-inal LENS condition (short are).
For the x/L = 0.1 station, the temperature and tangential velocity pro�les exhibit thecharacteristics of a merged layer. Farther downstream, the pro�les exhibit a distinctinviscid layer behind the leading edge shock. From the density pro�les, it is evident thatthe leading edge shock strength has reached an initial maximum near the x/L stationof 0.2 and continues to decrease in strength until the shock/shock interaction region isencountered.
The compression wavelets originating upstream of separation (Fig. 23) coalesce into ashock before the outer edge of the boundary layer is reached, with the de ection of theinviscid ow lagging behind that near the surface. Thus, the compression of the ow inthe boundary layer [15] varies from a somewhat gradual process, adjacent to the surface,to an oblique shock jump near the outer edge. The pressure pro�le at the x/L = 1.0station (Fig. 26) illustrates this behavior where the pressure behind the leading edgeshock is approximately two times the free-stream value but increases to approximately�ve times free stream behind the oblique separation shock.
4 CALCULATIONS FOR DOUBLE CONES
The results presented in this section are from ref. [8] where DSMC calculations weremade for both the ONERA R5Ch and the CUBRC LENS ow conditions for model con-�gurations that have been or will be tested. These results are included to highlight resultswhere the shock/shock interactions are much stronger than those previously discussed forthe hollow cylinder- are cases. For the double-cone models investigated, the �rst conehalf angle is 25�, while the second cone half angle is either 55� or 65�.
14
James N. Moss
4.1 25�/65� Cones at Mach 9.9 Air Flow|ONERA R5Ch Conditions
Calculations presented in this section are an extension of those reported in refs. [4]and [5], where the ow about a sharp double-cone model (25�/65�) with a maximumdiameter of 132.893 mm was calculated with both DSMC and Navier Stokes codes forReynolds numbers ranging from 24 719 to 247. This Reynolds number range was achievedby using the R5Ch nominal free-stream ow conditions (highest Reynolds number) andthen parametrically reducing the free-stream density. The current results are those forsmaller scale models with diameters between 66.4 and 121.0 mm. The maximum diameterof the current model con�guration that can be tested in the R5Ch wind tunnel is near 70mm, or about half the model size used in the previous studies. The con�guration is suchthat the lengths of the �rst and second cones are equal (L1 = L2, Fig. 29).
Figures 29 through 31 present representative results of the calculations. The natureof the shock interactions is demonstrated in Fig. 29 by using the �ne grid results forthe 66.4-mm-diameter model, a model size that the ONERA R5Ch wind tunnel shouldbe able to accommodate. Selected Mach contours are shown in which a large subsonicregion is located in front of the second cone. Locations for ow separation and reat-tachment along the surface are denoted by S and R, respectively. Also evident is thein uence of the separation shock on the oblique shock that is produced by the �rst cone,resulting in a triple point (T. P.) followed by a stronger transmitted shock that inter-acts with the stronger bow shock of the second cone|creating a second triple point.These shock/shock and shock/boundary layer interactions are induced by the larger coneand produce a signi�cant separation region characterized by a single vortex, indicatedby streamlines embedded within the subsonic region near the intersecting cones. Thegeneral ow structure evident for the 66.45-mm-diameter model is that found for thelarger 25�/65� models. However, when the model diameter is increased to 132.9 mm(results from ref. [5]), secondary vortices were evident, and a Navier-Stokes computationindicated some unsteadiness (small oscillations of separation location). Undoubtedly, the ow would become unsteady for larger models, as is demonstrated by the rapid growthof the extent of separation presented in Fig. 31.
Also included in Fig. 29 is information concerning the computational domain, whichconsisted of eight arbitrary regions. Each region is subdivided into cells, and the cellsin selected regions are subdivided into subcells to enhance the spatial resolution used toselect collision partners. Time step information for region one and the time interval forwhich the time-averaged results were obtained are included in Fig. 29.
Figure 30 presents the corresponding surface results|heating rate, pressure, and skinfriction|for the 66.45-mm-diameter model. The variable s denotes the model wettedlength measured from the cone vertex. Results are for a �ne-grid solution, a solution forwhich the surface results and extent of separation indicate grid independence [8]. Thequalitative features of the surface data are consistent with experimental measurements[15] for laminar separated ows. First, the separation position is in close agreement with
15
James N. Moss
the location of the �rst in ection point (maximum slope) of the initial pressure rise and thelocation where the heat transfer rate decreases signi�cantly with respect to the single coneresults. Second, the pressure reaches a plateau for a signi�cantly large separation region,while the heat transfer is signi�cantly reduced. Third, at or preceding the intersectionof the two cones, the heat transfer experiences a minimum and then increases rapidly, asdoes the pressure, with increasing distance along the second cone.
0 0.01 0.02 0.03 0.04 0.05x, m
0
0.01
0.02
0.03
0.04
0.05
y,m
Mach 9.91 air25o/65o double coned = 66.45 mmL1 = 25 mm
R
S
M = 9.0
12
3
4
5
7
6
8
Fine-grid results8-region domain31,410 cells464,180 subcells557,246 molecules∆t1 =12 nst1,avg = 2.16 to 3.15 ms
Region Cells Subcells/Cell1 35 x 40 3 x 22 55 x 75 10 x 23 125 x 85 12 x 24 20 x 50 1 x 15 20 x 80 1 x 16 90 x 100 4 x 27 40 x 90 4 x 38 3 x 40 4 x 2
M = 1.0
xS = 13.86 mmxR = 25.95 mm∆s /L1 = 0.700
L1
L2
T. P.
T. P.
0.0 0.5 1.0 1.5 2.0s/L1
0
20
40
60
80
100
120
140
q,kW
/m2
0
200
400
600
800
1000
1200
1400
1600
τan
dp,
N/m
2
Mach 9.91 air25o/65o double coned = 66.45 mmRe∞,d = 12,360
s1 = 25 mms2 = 50 mmL1 = 25 mmFine-grid results
p
τq
sS/L1 = 0.612sR/L1 = 1.312
Figure 29: Flow structure and simulation param-eters 25�=65� double cone (R5Ch ow).
Figure 30: Calculated surface results for double-cone model at R5Ch ow conditions.
102 103 104
Re∞,d
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
s/L 1
and
∆s/L
1
Circles-Separation location, s/L1Deltas-Reattachment location, s/L1Squares--Separation length, ∆s/L1Filled--Ref. 5, density variableOpen--Ref. 8, model size variable,
coarse grid
Mach 9.91 air25o/65o double coneL1 = L2
0.0 0.5 1.0 1.5 2.0 2.5s/L1
0
250
500
750
1000
1250
1500
q,kW
/m2
SeparationGrid Cells Subcells ∆s/L1VC 39,429 39,429 0.186C 81,240 81,240 0.246I 81,240 396,240 0.269
L1 = 101.59 mm
Mach 9.25 nitrogen25o/ 55o double coned = 261.8 mmRe∞,d = 42,150
V∞ = 2,657.9 m/sρ∞ = 7.78 x 10-4 kg/m3
T∞ = 198.9 KTW = 293 K
Figure 31: Separation data as a function ofReynolds number for R5Ch ow conditions.
Figure 32: E�ect of grid on heating-rate resultsfor a pretest nominal LENS ow condition.
The sensitivity of the extent of separation to Reynolds number is presented in Fig. 31,where the results are a summary of those presented in ref. [5] (constant model diameter
16
James N. Moss
of 132.9 mm and varying the free-stream density) and ref. [8] (the R5Ch ow conditionsand varying the model diameter). The two data sets show a consistent trend: the extentof separation increasing with Reynolds number and a very rapid increase in separationnear a Reynolds number of 25 000. For the conditions investigated, the calculations showthat separation persists for Reynolds numbers as low as about 800.
4.2 25�/55� Cones at Mach 9.6 Nitrogen Flow|CUBRC LENS Conditions
This section focuses on results for the CUBRC LENS ow conditions where the maxi-mum model diameter is 261.8 mm. The initial calculations were made for pretest nominalconditions (included in Fig. 32). A grid sensitivity study was conducted, and the resultsof this study, as it impacts the extent of separation (�s/L1) and surface heating-ratedistributions, are presented in Fig. 32. The qualitative characterization of the grid listedin Fig. 32 is I for intermediate, C for coarse, and VC for very coarse. The �nest gridused in this exercise is described as intermediate because grid independence was notdemonstrated|additional grid re�nement is necessary to indicate whether the currentintermediate grid is adequate.
0.00 0.05 0.10 0.15 0.20x, m
0.00
0.05
0.10
0.15
0.20
y,m
12
3
4
5
6
7
8
Intermediate-grid results8-Region domain81,240 cells396,240 subcells1,381,621 molecules∆t1 = 2 nst1,avg = 1.12 to 1.35 ms
Mach 9.56 nitrogen25o/55o double cone
d = 261.8 mmRe∞,d = 38,340L1 = 101.59 mmL2 = 107.41 mm
M = 9.0
M= 1.0
S
RxS = 80.02 mmxR = 99.87 mm∆s/L1 = 0.265
Region Cells Subcells/Cell1 50 x 160 1 x 12 40 x 240 3 x 13 45 x 360 6 x 24 80 x 240 2 x 25 20 x 200 1 x 16 20 x 200 1 x 17 100 x 200 2 x 28 6 x 40 1 x 1
0.0 0.5 1.0 1.5 2.0 2.5s/L1
0
200
400
600
800
1000
1200
1400
q,kW
/m2
0
2000
4000
6000
8000
10000
12000
pan
dτ,
N/m
2pq5 x τ
Mach 9.56 nitrogen25o/55o double coned = 261.8 mmRe∞,d = 38,340L1 = 101.59 mmL2 = 107.41 mmIntermediate-gridresults
V∞ = 2713.6 m/sρ∞ = 6.808 x 10-4 kg/m3
T∞ = 194.1 KTW = 297.8 K
sS/L1 = 0.869sR/L1 = 1.134∆s/L1 = 0.265
Figure 33: Flow structure and simulation param-eters for 25�=55� double cone (LENS ow).
Figure 34: Calculated surface results for double-cone model at a LENS test condition.
Once information became available as to the actual test conditions used in the CUBRCexperiments, the �nest grid used in the pretest grid investigation was then used to makea calculation for the actual LENS test conditions (free-stream conditions included in Fig.34), and selected results are presented in Figs. 33 and 34. The general features of theshock layer structure are given in Fig. 33 where selected Mach contours are included, alongwith details of the numerical parameters used in the simulation. Values for the surfacequantities are shown in Fig. 34. For the surface pressure distribution, the calculatedvalues outside the region in uenced by the shock/boundary layer interactions are in close
17
James N. Moss
agreement with the inviscid cone values (ref. [17]) of 948 N/m2 along the 25� cone and3710 N/m2 along the 55� cone. Details concerning the ow structure are presented inref. [8] where the calculations show that the maximum values for density and scalarpressure are 155 and 319 times their respective free-stream values and that there is amaximum overall kinetic temperature of 3104 K. Comparisons of translational and internaltemperature pro�les show that the nonequilibrium e�ects are con�ned primarily to theouter bow shock crossings. Opportunities will exist for comparing the present results withthe experimental measurements (heating-rate and pressure distributions) that have beencompleted when the CUBRC data are released.
5 CONCLUDING REMARKS
Results of a computational study are presented for Mach 10 ow about hollow cylinder are and sharp double cone models where the combination of model con�gurations, size,and ow conditions produce a signi�cant range of shock/shock and shock/boundary layerinteractions. The computations are made with the direct simulation Monte Carlo (DSMC)method, hence, low Reynolds number ows. The results presented provide insight into thenature of the shock interactions, their impact on surface quantities, and the sensitivityof the results to computational parameters for ow conditions that can be produced incurrent ground-based facilities.
Results of the hollow cylinder- are calculations are compared with the experimentalsurface measurements made in the ONERA R5Ch wind tunnel (Mach 9.91 air at a Re1;L
= 18 916). The extent of the calculated separation region is very sensitive to the gridresolution used|a coarse grid results in a smaller separation region. Results for the�nest grid investigated show very good agreement with the experimental measurementsfor the separation and reattachment locations and surface heating. For surface pressure,the agreement between calculation and measurement is poor|the calculated values areuniformly high along both the cylinder and are by a factor of 1.4. Additional DSMCcalculations are made for the cylinder to examine the impact of additional re�nementof solution parameters and leading edge treatment, and the results show no signi�cantimpact on the previously reported results. Based on these �ndings, it is believed theDSMC results for surface pressure are correct along the cylinder and should be reasonablyaccurate for the are.
Also, DSMC results are presented for two hollow cylinder- are models (one having thesame outer surface as the ONERA model and one with a longer are) that will be testedin the LENS facility at a proposed nominal test condition (Mach 9.57 nitrogen at a Re1;L
= 14 920). Information concerning the e�ect of grid resolution is presented along withdetailed data concerning surface results and ow structure. For the LENS test case, theextent of separation is much smaller than that calculated and measured for the ONERAtests, and the calculated surface results are essentially identical for the long and short are models.
For the double cone models, the extent of separation as a function of free-stream
18
James N. Moss
Reynolds number is demonstrated for hypersonic cold ow conditions, and the currentresults are shown to be consistent with previous calculations. These results are for theONERA R5Ch wind tunnel ow conditions and 25�/65� double-cone models. Calculatedresults are presented for free-stream Reynolds numbers (based on maximum body diam-eter) of 247 to 24 719. Preliminary tests in the R5Ch tunnel have demonstrated thatexperiments can be conducted on this double-cone con�guration for Reynolds numbers aslarge as 12 000. Computations were also made for a higher enthalpy test conducted in theCUBRC LENS impulse facility with a much larger diameter model and a 25�/55� double-cone con�guration. Results of a grid sensitivity investigation are discussed, and surfaceresults are presented for a test condition that has been conducted (Mach 9.56 nitrogenat a Re1;L = 38 340). Opportunities should exist for comparing the current results withexperimental measurements for surface heating and pressure distributions.
6 ACKNOWLEDGMENTS
The author wishes to acknowledge the assistance of B. Chanetz and T. Pot of ONERA,Chalais-Meudon, for providing detailed information regarding their experiments and theresults of their measurements. Also, the author wishes to acknowledge the assistance ofMichael Holden of CUBRC for providing information regarding the model con�gurationsand ow conditions used in the LENS tests.
REFERENCES
[1] Bird, G. A.; Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Ox-ford: Clarendon Press, 1994.
[2] Moss, J. N., Dogra, V. K., and Price, J. M.; DSMC Simulations for a Hollow Cylinder-Flare Con�guration. AIAA Paper 94-2015, June 1994.
[3] Moss, J. N., Dogra, V. K., Price, J. M., and Hash, D. B.; Comparison of DSMC andExperimental Results for Hypersonic External Flows. AIAA Paper 95-2028, June1995.
[4] Moss, J. N., and Olejniczak, J.; Shock-Wave/Boundary-Layer Interactions in Hyper-sonic Low Density Flows. AIAA Paper 98-2668, 1998.
[5] Moss, J. N., Olejniczak, J., Chanetz, B., and Pot, T.; Hypersonic Separated Flowsat Low Reynolds Number Conditions. Proceedings of the 21st International Sympo-sium on Rare�ed Gas Dynamics, Brun, Campargue, Gatignol, and Lengrand, eds.,Cepadues-Editions, Toulouse, France, Vol. II, 1999, pp. 617-624.
[6] Chanetz, B., Benay, R., Bousquet, J.-M., Bur, R., Pot, T., Grasso, F. and Moss, J.;Experimental and Numerical Study of the Laminar Separation in Hypersonic Flow.Aerospace Science and Technology, No. 3, 1998, pp. 205-218.
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James N. Moss
[7] Chanetz, B., Bur, R., Pot, T., Pigache, D., Grasso, F., and Moss, J.; Experimentaland Numerical Study of the Laminar Separation in Hypersonic Flow. Paper presentedat ECCOMAS 2000, Barcelona, Spain, Sept. 11-14, 2000.
[8] Moss, J. N.; DSMC Simulations of Shock Interactions About Sharp Double Cones.presented at the 22nd International Symposium on Rare�ed Gas Dynamics, Sydney,Australia, July 9-14, 2000. Available as NASA TM-2000-210318, August 2000.
[9] Bird, G. A.; The G2/A3 Program System Users Manual. Version 1.8, March 1992.
[10] Borgnakke, C. and Larsen, P. S.; Statistical Collision Model for Monte Carlo Simu-lation of Polyatomic Gas Mixture. Journal of Computational Physics, Vol. 18, No. 4,1975, pp. 405-420.
[11] Hypersonic Experimental and Computational Capability, Improvement and Valida-tion. AGARD AR 319, edited by J. Muylaert, A. Kumar, and C Dujarric, Vol. II,Dec. 1998.
[12] Candler, G. V., Nompelis, I., and Holden, M. S.; Computational Analysis of Hyper-sonic Laminar Viscous-Inviscid Interactions. AIAA Paper 2000-0532, Jan. 2000.
[13] Holden, M.; Experimental Studies of Laminar Separated Flows Induced by ShockWave/Boundary Layer and Shock/Shock Interaction in Hypersonic Flows for CFDValidation. AIAA Paper 2000-0930, Jan. 2000.
[14] Chanetz, B.; Study of Axisymmetric Shock Wave/Boundary Layer Interaction inHypersonic Laminar Flow. ONERA Technical Report TR No. 42/4623, Feb. 1995.
[15] Needham, D., and Stollery, J.; Boundary Layer Separation in Hypersonic Flow. AIAAPaper 66-455, 1966.
[16] Markelov, G. N., Kudryavtsev, A. N., and Ivanov, M. S.; Continuum and KineticSimulation of Laminar Separated Flow at Hypersonic Speeds. Journal of Spacecraftand Rockets, Vol. 37, No. 4, July-August 2000, pp. 499-506.
[17] Ames Research Sta�; Equations, Tables, and Charts for Compressible Flow. NACAReport 1135, 1953.
20