Research Institute for Advanced Computer ScienceNASA Ames Research Center

NASA-CR-199754

An Edge-Based Solution-Adaptive MethodApplied to the AIRPLANE Code

Rupak BiswasScott D. Thomas

Susan E. Cliff

(NASA-CR-199754) AN EDGE-BASEDSOLUTION-ADAPTIVE METHOD APPLIED TOTHE AIRPLANE CODE (Research Inst.for Advanced Computer Science)13 p

N96-15350

Unclas

G3/34 0081590

RIACS Technical Report 95.22 November 1995

Paper No. AIAA-96-0553, presented at the AIAA 34th Aerospace Sciences Meeting & Exhibit,Reno, Nevada, January 15-18, 1996

https://ntrs.nasa.gov/search.jsp?R=19960008184 2020-02-10T05:49:48+00:00Z

An Edge-Based Solution-Adaptive MethodApplied to the AIRPLANE Code

Rupak BiswasScott D. Thomas

Susan E. Cliff

The Research Institute of Advanced Computer Science is operated by Universities Space ResearchAssociation, The American City Building, Suite 212, Columbia, MD 21044, (410) 730-2656

Work reported herein was supported by NASA via Contract NAS 2-13721 between NASA and the UniversitiesSpace Research Association (USRA). Work was performed at the Research Institute for Advanced ComputerScience (RIACS), NASA Ames Research Center, Moffett Field, CA 94035-1000.

AN EDGE-BASED SOLUTION-ADAPTIVE METHOD

APPLIED TO THE AIRPLANE CODE*

Rupak Biswas^RIACS, NASA Ames Research Center, Moffett Field, CA

Scott D. Thomas*Sterling Software Inc., 1121 San Antonio Road, Palo Alto, CA

Susan E. Cliff5

High Speed Aerodynamics Branch, NASA Ames Research Center, Moffett Field, CA

ABSTRACT

Computational methods to solve large-scale re-alistic problems in fluid flow can be made more effi-cient and cost effective by using them in conjunctionwith dynamic mesh adaption procedures that per-form simultaneous coarsening and refinement to cap-ture flow features of interest. This work couples thetetrahedral mesh adaption scheme, called 3D.TAG,with the AIRPLANE code to solve complete aircraftconfiguration problems in transonic and supersonicflow regimes. Results indicate that the near-fieldsonic boom pressure signature of a cone-cylinder isimproved, the oblique and normal shocks are bet-ter resolved on a transonic wing, and the bow shockahead of an unstarted inlet is better defined.

INTRODUCTION

Traditional computational methods can bemade more efficient and cost effective by redistribut-ing the available mesh points to capture flowfieldphenomena of interest. Such adaptive proceduresevolve with the solution and provide a robust andreliable methodology. Highly localized regions ofmesh refinement are required in order to accuratelycapture shock waves, contact discontinuities, andshear layers. This provides the aerodynamicist withthe opportunity to obtain flow solutions on adaptedmeshes that are comparable to those obtained onglobally-fine grids.

Two types of solution-adaptive grid strategiesare commonly used with unstructured-grid methods.

'This paper is declared a work of the U.S. Government andis not subject to copyright protection in the United States.

tScientist, Member AIAA.'Software Specialist, Senior Member AIAA.5 Aerospace Engineer, Member AIAA.

Grid regeneration schemes generate a new grid witha higher or lower concentration of points in regionsthat are targeted by some error indicator. A ma-jor disadvantage of such schemes is that they arecomputationally expensive. This is a serious draw-back for unsteady problems where the mesh mustbe frequently adapted. However, resulting grids areusually well-formed with smooth transitions betweenregions of coarse and fine mesh spacing.

Local mesh adaption, on the other hand, in-volves adding points to the existing grid in regionswhere the error indicator is high, and removingpoints from regions where the indicator is low. Theadvantage of such strategies is that relatively fewmesh points need to be added or deleted at eachadaptive step for unsteady problems. However, com-plicated logic and data structures are required tokeep track of the points that are added and removed.Furthermore, the resulting grids can often have non-smooth transitions between coarse and fine mesh re-gions and a criterion for maintaining a level of gridquality is usually required.

This work couples the dynamic tetrahedralmesh adaption scheme,1 called 3D.TAG, with theAIRPLANE code2'3 to solve complete airplane con-figuration problems in transonic and supersonic flowregimes. The objective is to demonstrate the effec-tiveness of the local mesh adaption method to obtainimproved solutions.

3D.TAG has been previously combined withTRI3D,4 a finite volume upwind Euler code, to suc-cessfully model large problems in helicopter aerody-namics and aeroacoustics.5'6 Results have demon-strated that appropriate error indicators can ef-ficiently capture surface shocks, propagate acous-tic signals with minimal dissipation, and accuratelyconvect rotorcraft vortical wakes.

The remainder of this report is divided intofive sections. The first section briefly describes theAIRPLANE code. The next section describes the3D_TAG tetrahedral adaption procedure. The thirdsection highlights the modifications that were nec-essary in order to interface the two codes. Resultsobtained for a supersonic cone-cylinder, a transonicwing, and a supersonic inlet unstart problem are re-ported in the fourth section. Finally, we concludeby summarizing our observations from this work.

EULER FLOW SOLVER

AIRPLANE consists of two separate codes:FLOPLANE and MESHPLANE — the Euler flowsolver and the grid generator, respectively. Theseprograms were developed by Jameson and Bakerto model complex configurations using unstructuredgrids. FLOPLANE3 uses a finite-volume algorithmthat computes flow variables at the vertices of atetrahedral mesh. MESHPLANE7'8 generates thetetrahedral elements by using a constrained Delau-nay triangulation algorithm. The often complicatedand time-consuming procedure of blocking and grid-ding structured multi-zone grids is eliminated byAIRPLANE, but some control over the distributionof mesh points is lost in the process.

In order to improve the turn-around time andcomputational efficiency, a parallel version of AIR-PLANE has been recently established on the IBMSP2.9 Results show that almost perfect scalability isobtained for up to 64 processors. This parallel ver-sion of AIRPLANE is used for the calculations inthis report.

TETRAHEDRAL ADAPTION SCHEME

3D_TAG has its data structure based on edgesthat connect the vertices of a tetrahedral mesh. Thismeans that each tetrahedral element is defined byits six edges rather than by its four vertices. Thisedge data structure makes the mesh adaption pro-cedure capable of performing anisotropic refinementand coarsening. A successful data structure mustcontain only the information required to rapidly re-construct the mesh connectivity when vertices areadded or deleted while having a reasonable memoryrequirement.

At each mesh adaption step, individual edgesare marked for coarsening, refinement, or no change.Only three subdivision types are allowed for eachtetrahedral element and these are shown in Fig. 1.The 1:8 isotropic subdivision is implemented by

adding a new vertex at the mid-point of each of thesix edges. The 1:4 and 1:2 subdivisions can resulteither because the edges of a parent tetrahedron aretargeted anisotropically or because they are requiredto form a valid connectivity for the new mesh. Whenan edge is bisected, the solution vector is linearly in-terpolated at the mid-point from the two points thatconstitute the original edge.

1:4 1:2

Figure 1: Three types of subdivision are permittedfor a tetrahedral element.

Mesh refinement is performed by first setting abit flag to one for each edge that is targeted for sub-division. The edge markings for each element arethen combined to form a binary pattern as shown inFig. 2 where the edges marked with an R are the onesto be bisected. Once this edge-marking is completed,each element is independently subdivided based onits binary pattern. Special data structures are usedin order to ensure that this process is computation-ally efficient.

6 5 4 3 2 1

0 0 1 0 1 1

Edge#

Pattern - 1 1

R

Figure 2: Sample edge-marking pattern for elementsubdivision.

Mesh coarsening also uses the edge-markingpatterns. If a child element has any edge marked forcoarsening, this element and its siblings are removedand their parent element is reinstated. The parentedges and elements are retained at each refinementstep so they do not have to be reconstructed. Rein-stated parent elements have their edge-marking pat-terns adjusted to reflect that some edges have beencoarsened. The mesh refinement procedure is theninvoked to generate a valid mesh.

A significant feature in 3D_TAG is the concept

of "sublists." A data structure is maintained whereeach vertex has a sublist of all the edges that areincident upon it. Also, each edge has a sublist of allthe elements that share it. These sublists eliminateextensive searches and are crucial to the efficiency ofthe overall adaption scheme.

In the initial version of the code,1 the datastructure was implemented in C as a series ofdynamically-allocated linked lists. This facilitatedthe addition and deletion of mesh points, but thelinked lists made it very difficult to pass informationdirectly to Fortran flow solvers. In order to reducethe communication overhead, the linked lists havebeen replaced with arrays and a garbage collectionalgorithm is used to compact free space when meshpoints are removed.

An important component of any mesh adaptionprocedure is the choice of an error indicator for eachregion of the mesh. Typically, this error indicatoris not a true estimate of the error in the solution;rather, it is an indicator of high gradients in theflowfield that are assumed to be regions of high er-ror. Error indicators are usually problem-dependentand considerable fine tuning is often necessary to ob-tain satisfactory results.

COUPLING AIRPLANE AND 3D.TAG

The AIRPLANE and the 3D_TAG codes arecurrently loosely coupled. Relevant information ispassed between the two programs via files. Histori-cally, FLOPLANE expected the computational gridto be provided by MESHPLANE. With the inclusionof the mesh adaption procedure, the flow solver mustalso be able to read intermediate adapted meshesas well as read and write solution files generatedby 3D_TAG. The input module of FLOPLANE hasbeen modified to make this possible. 3D.TAG hasalso been customized to read and write mesh filesthat are in the format generated by MESHPLANEand expected by FLOPLANE.

Figure 3 shows how the three codes inter-act. MESHPLANE generates an initial unstruc-tured tetrahedral grid based on input mesh param-eters. This mesh is passed on to FLOPLANE whichcomputes an acceptably converged solution. The ini-tial mesh is also read by 3D.TAG that converts it toan edge-based representation and generates all thesublists.

When the first mesh refinement is desired,3D.TAG is invoked and supplied with the initial con-verged solution. 3D_TAG generates a new mesh andinterpolates the coarse-grid solution onto the new

Initial Mesh Parameters

1st Adaption

Edge-basedAdapted Grid

2nd Adaption

Figure 3: Interfacing AIRPLANE and 3D.TAG viafiles.

mesh. This mesh and the interpolated solution aregiven as input to FLOPLANE to further convergethe solution. This entire process is then repeated iffurther mesh adaption steps are desired.

Note that at the end of each adaption step,3D_TAG writes out its entire internal data structureto a file. This file not only contains all the meshconnectivity but also all information pertaining tothe history of the mesh. The overhead of retainingstorage for the parent elements and edges is typi-cally small (less than 15% for the test cases in thisreport) but required to coarsen the mesh and/or toimprove grid quality.10

The coupling strategy allows both AIRPLANEand 3D.TAG to remain modular and independent.However, because the interfacing is done throughfiles, human intervention is required whenever themesh has to be adapted. This is not a drawbackfor steady-flow problems where the mesh has to be

adapted only a few times and the user needs to visu-alize the grid and the solution before deciding howand where to adapt the mesh. Note that the en-tire procedure of repeatedly adapting the mesh andrunning the flow solver can be easily automated forproblems with unsteady flows.

COMPUTATIONAL RESULTS

The coupled 3D.TAG and AIRPLANE codeswere applied to a supersonic cone-cylinder, a tran-sonic wing, and a supersonic inlet unstart problem.These examples were chosen as test cases that fullyexercised all aspects of the mesh adaption scheme inthree dimensions as well as demonstrated the rangeof problems that could be efficiently solved using thecombined method.

The first test case involves computation of thenear-field sonic boom pressure signature for a cone-cylinder. The cone-cylinder consists of three com-ponents: a cone of length L — 8.6 inches with a6.48° included angle, a cylinder of length 1L extend-ing downstream of the cone, and a second cone toclose the cylinder to a point at a distance 4L fromthe nose. One-half of the configuration was modeledwith AIRPLANE, since axisymmetric and quarter-plane modeling are not permitted. Flow conditionsfor this case were Mach 1.68 and an angle of attackof zero degrees.

A very coarse starting mesh was desired torapidly assess the AIRPLANE-3D-TAG couplingwith minimal computational resources. MESH-PLANE was not used to generate the initial compu-tational mesh because without modification to thealgorithm, it is unable to generate extremely coarsemeshes. MESHPLANE was developed to automat-ically generate meshes for complete aircraft. As aresult, it generates grids of sufficient density nearthe surface to obtain reliable results.

The initial three-dimensional unstructuredmesh for this case was created by subdividing amulti-block structured grid. Most hexahedra of athree-block C-H grid were subdivided into five tetra-hedra. The near-, mid-, and far-field grid density ofthe three blocks decreased with distance from themodel. The surface geometry was defined at 65 sta-tions distributed uniformly from nose to tail. Cir-cumferential cuts had 17 points, except the nose andtail were defined by a single point. The entire gridwas swept at the free-stream Mach angle to reducethe number of mesh points. The upstream boundarywas 0.5L ahead of the nose, while the downstreamboundary was at least 6.7L aft of the tail. The up-

per, lower, and side boundaries were greater than15L from the axis of the body.

The initial mesh consisted of 19,957 points, ofwhich only 1073 were on the surface of the body. Thesmall number of surface points led to faceting thatwas somewhat reduced by using the 3D_TAG code togeometrically refine the mesh within a neighborhoodfrom the upstream boundary to the midpoint of thecylinder with radius 2.5 times that of the body.

The purpose of this test case is to predict thesonic boom pressure signature below the body andcompare it with experimental data. The computa-tional pressure signature was obtained at a distanceL below the configuration (h/L — 1) in the sym-metry plane and extrapolated to the experimentaldistance using the waveform parameter method ofThomas.11 Since we are primarily interested in theweak finite-rise shock from the nose of the cone andthe expansion waves from the cone-cylinder junction(shoulder), mesh refinement was restricted to a dis-tance of 2L from the body in the radial directionand a distance 3L downstream of the nose, shearedat the free-stream Mach angle.

Table 1: Grid sizes through the adaption steps

Initial meshGeom. ref.1st adapt.2nd adapt.3rd adapt.

Vertices19,95728,82171,785

129,722212,251

Elements88,216

132,776367,522703,396

1,187,366

Edges112,691168,154451,924848,171

1,415,967

A total of three solution-adaptive steps wereperformed. An error indicator based on pressuredifferences across edges was used to adapt the mesh.However, since acoustic pressure attenuates with dis-tance from the source, the pressure difference acrossan edge was scaled by a function proportional to itsdistance from the body to obtain the error indicator.Converged solutions were obtained with 2000 FLO-PLANE iterations after each refinement step. Ta-ble 1 presents the progression of grid sizes throughthe four adaption steps.

Figure 4 shows the sequence of meshes in thesymmetry plane. The initial mesh as well as theadapted meshes after the three solution-adaptivesteps are shown. Note that a coarsening phase re-moved some surface points on the cylindrical portionof the model where the pressure gradients are smallor non-existant. Also, each refinement step was per-formed to maintain the same point density in thevicinity of the shock to at least a distance L from

Figure 4: Several mesh adaption stages for the cone-cylinder, MOO = 1-68, a = 0.0°.

the body where the pressure was sampled. A closeupof the final adapted mesh, shown in Fig. 5, confirmsthis. Horizontal lines are placed at the sampling lo-cation. Pressure contours are also depicted in Fig. 5.These contours show excellent resolution of the ex-pansion waves at the shoulder.

The AIRPLANE and experimental pressure sig-natures are compared at an h/L of 10.0 in Fig. 6.Results are shown for both the initial coarse and thefinal adapted meshes. The computational data weretaken at h/L = 1.0 and extrapolated to the experi-mental distance. For the sake of continuity, resultsare also shown from earlier work12 where the compu-

tational data was extrapolated from an h/L of 0.1.The earlier computation underpredicts the finite-riseportion of the signature and the maximum overpres-sure. The new results on the final adapted meshmore accurately predict the finite-rise bow shockand maximum overpressure, but the expansion isslightly underpredicted in spite of the grid refine-ment around the shoulder. The pressure signaturesfor the adapted grid can be taken at larger distancesfrom the model than for the non-adapted grid. Thisis significant since accurate signatures cannot be ob-tained at small sampling distances from typical non-axisymmetric configurations. The three-dimensionaleffects require a sampling distance of approximatelyhalf the span for reliable data. Finally, comparisonwith the coarse mesh results demonstrate the effec-tiveness of mesh adaption.

The second test case consists of a flow-field com-putation for the ONERA M6 wing13 at Mach 0.84and 3.06-degrees angle of attack . The model wasan isolated, swept, tapered, untwisted wing mountedon a wind-tunnel wall. All wing sections are scaledfrom the 9.8%-thick symmetric root section. The tipchord is 0.562 and the half-span is 1.48, based on aroot chord of unit length. The half-chord sweep ofthe wing is 23.2°. The primary challenge here is toaccurately capture the lambda shock on the uppersurface of the wing. The oblique shock near the lead-ing edge is much weaker than the normal shock andis generally more difficult to resolve.

The initial three-dimensional computationalmesh was created from a structured H-H mesh usinga simple algebraic method. The outer boundarieswere located approximately 15 root chords awayfrom the wing in all directions. Each structured-gridhexahedron was then split into at most five tetrahe-dra. Collapsed hexahedra filling the wedge regionbeyond the flat tip yielded fewer than five tetrahe-dra each. Careful selection of diagonal edges at theblunt wing leading edge prevented the possibility ofrefined elements penetrating the surface.

The initial unstructured mesh consisted of34,202 points, of which only 103 were on the wingsurface. The small number of surface points led tosevere faceting of the wing surface. Before a solutionwas attempted, this undesirable faceting was some-what diminished by using the 3D.TAG code to geo-metrically refine the mesh within a box enclosing theM6 wing. This box extended one-half root chordsfore and aft, above and below, and spanwise beyondthe tip of the wing, and was swept with the leadingand trailing edges. This refinement step increasedthe number of surface points to 378 and also reduced

Figure 5: Close-up view of the final adapted mesh and pressure contours on the symmetry plane for thecone-cylinder, M^, = 1.68, a = 0.0°.

8.0

I

4.0

0.0

-4.0-

-8.0

o Experimental data— FromRef. 12

Coarse initial mesh— Final adapted mesh

0.0 0.5 1.0x/L

1.5 2.0

Figure 6: Pressure signature for the cone-cylinder,Moo = 1.68, a = 0.0°, h/L = 10.0.

the likelihood of important flow features from beingcompletely missed.

A total of four solution-adaptive steps were thenperformed. Innovative error indicators that includespecial shock finders have been used in earlier work14

to adapt the computational mesh; however, using

the absolute value of the pressure difference acrossan edge as a measure of the error worked remarkablywell. This decision criterion was very effective incapturing flow gradients at the leading edge, in theupper surface expansion, and at the shocks. Theresidual was reduced by at least three orders on eachintermediate mesh using FLOPLANE.

Table 2 presents the progression of grid sizesthrough the five adaption steps. The final adaptionwas prevented from refining edges that lay within acylindrical region of radius 0.04 centered around theleading edge stretching from the symmetry plane tothe outer spanwise boundary. The leading edge re-gion was already adequately resolved since the pres-sure gradients are large and the adaption method re-fines this region first. New points were added alongthe surface shocks in the final refinement step in anattempt to accurately capture the shocks. The finalmesh therefore has four levels of refinement near theleading edge but five levels along the shocks.

For the purposes of critically evaluating theadapted mesh and solution, FLOPLANE was alsorun on a uniformly fine mesh (224,354 points,1,352,104 elements, and 1,598,508 edges) generated

Figure 7: Finest MESHPLANE and final adapted meshes and corresponding pressure contours on the uppersurface of the ONERA M6 wing, Mx = 0.84, a = 3.06°.

Table 2: Grid sizes through the adaption steps

Vertices Elements EdgesInitial mesh 34,202 153,706 194,657Geom. ref.1st adapt.2nd adapt.3rd adapt.4th adapt.

40,03687,058

153,326227,433325,901

185,232455,060834,834

1,260,0691,830,957

232,610551,795

1,001,9251,505,0582,176,617

by MESHPLANE.Figure 7 shows the MESHPLANE and the final

solution-adapted meshes along with the correspond-ing pressure contours on the upper surface of thewing. Note that the resolution is significantly im-

proved along the shocks for the adapted mesh eventhough it has only 9802 surface points compared to11,718 for the MESHPLANE mesh. It is obviousthat the adapted mesh with fewer points has resultedin more distinct shocks.

During each refinement step, a previously-generated database of grid points defining the wingsurface with very high resolution was interrogatedto ensure that all new surface points lie on the wing.This was required because the regular edge-midpointrefinement scheme would have preserved the initialfaceting of the wing surface.

Computed pressure coefficients on the finaladapted mesh at four span locations are presentedin Fig. 8. These results are compared to exper-imental data collected by Schmitt and Charpin.13

1.5

1.0

0.5

u

o.o-

-0.5-

-I.O-'T

44% 65%

1.5

1.0

0.5-

u

0.0

-0.5-

80% 90%

» Experiment (upper)T Experiment (lower)

MESHPLANE mesh.... Final 3D_TAG mesh— MESHPLANE mesh with adaption

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

X/C X/C

Figure 8: Comparison of surface pressure distributions at 44%-, 65%-, 80%-, and 90%-span stations for theONERA M6 wing, M^ = 0.84, a = 3.06°.

The experimental data is shown only as a reference,since the inviscid Euler solutions will not agree withthe data. These results are also compared to re-sults obtained on the globally fine mesh generated byMESHPLANE, and results obtained after one adap-tion of this mesh. Adapting the mesh from MESH-PLANE would be the typical method for improv-ing the solution. Fewer refinement levels should benecessary when starting from a uniformly fine mesh,which should reduce the chances of developing tetra-hedra of poor quality when subdivided.

The solution obtained with the mesh adaptedfrom the coarse structured grid shows better shockdefinition than the other computations and capturesboth shocks. It is well known that an accurate Eu-

ler solution will predict shocks that are stronger andaft of experiment, as is the case with these compu-tations. It is not possible to determine whether thissolution represents the most accurate Euler resultof the three solutions shown, since the strength andlocation of the shocks for an accurate inviscid resultare not known.

The results for the non-adapted mesh have lesscrisp shocks, and the forward shock at these sta-tions is not adequately captured. The results of oneadaption of the MESHPLANE grid more accuratelycaptures the forward shock shown in the three in-board span stations, but the the shocks are not aswell defined as the shocks obtained from the coarsestructured grid. It would be expected that further

Figure 9: Close-up view of the final adapted mesh and pressure contours on the lower surface of the supersonictransport configuration.

grid refinement steps would lead to more sharply de-fined shocks.

The results for one adaption of the standardmesh and the non-adapted grids do correlate bet-ter with experiment, but this is due to the artificialviscosity in the numerical scheme mimicing the ac-tual viscosity. The reduced cell size in the vicinity ofthe shocks associated with the adapted meshes, ledto pre- and post- shock pressure fluctuations, whichwere damped by increasing the local dissipation pa-rameter in FLOPLANE.

The final test problem is a complete supersonictransport configuration including nacelles. Only halfthe aircraft needs to be modeled because of sym-metry. The outboard nacelle is plugged to simu-late inlet unstart. This causes a bow shock to beformed upstream of the plugged nacelle at super-sonic speeds. This shock impinges on the lower sur-face of the wing and also reflects off the inboard na-celle. Adaptive mesh refinement using the 3D.TAGcode was performed in this region to accurately cap-

ture the shock and help understand more of the flowphysics.

The initial MESHPLANE mesh consisted of457,712 grid points and over 2.75 million tetrahe-dral elements. More than 40,000 iterations were re-quired to produce a well-converged solution. Onemesh adaption was then performed within a rectan-gular region enclosing the bow shock using pressuredifferences across edges as the error indicator. Thisincreased the mesh size to 497,211 points and about3 million tetrahedra.

The final mesh and pressure contours on thelower surface of the configuration in the vicinity ofthe nacelles are shown in Fig. 9. Only triangleswith visible centroids are drawn in order to approx-imately hide triangles behind the nacelles. In theinitial mesh, stream wise mesh density was doubledin a rectangular region of the wing surface near theinlet, but the bow shock was upstream of this region.3D.TAG refined much of the rectangular region andalso elements ahead of it, where the pressure differ-

Figure 10: Pressure contours on a planar slice through the axis of the plugged nacelle before and after meshrefinement.

ences along edges were large. There is refinementnear the leading edge of the plugged inlet and wherethe bow shock strikes the inboard nacelle. The in-let of the inboard nacelle was not refined becauseair flows supersonically through it with a relativelysmall pressure gradient.

Meshes and pressure contours are shown ona vertical cutting plane aligned with the axis ofthe plugged nacelle for the initial and final meshesin Fig. 10. The slices are composed of trianglesand quadrilaterals. Pressure values are obtained bylinear interpolation from the node-based solutionson the tetrahedral meshes. Refinement follows thestrongest portion of the bow shock and is swept in afashion similar to that observed on the surface. The

width of the shock is roughly cut in half. Pressuredifferences are very high at the lip of the inlet, bothinside and out, and in the gap between the inletleading edge and the lower wing surface.

Figure 11 presents a qualitative comparison ofthe pressure ratio along the axis forward of theplugged nacelle on the initial and refined meshes.As noted above for Fig. 10, the shock on the re-fined mesh is half as wide as on the initial mesh.Pre- and post-shock oscillations are also confined toa narrower region on the refined mesh. The pres-sure ratio ahead of the shock is the same for bothmeshes, but the predicted change in pressure ratiois 4.5% higher for the refined mesh.

10

— Coarse initial mesh— Final adapted mesh

Figure 11: Variation of the pressure ratio along theaxis of the plugged nacelle.

SUMMARY

The tetrahedral mesh adaption scheme was suc-cessfully coupled to the AIRPLANE code. Thenear-field sonic boom pressure signature on a cone-cylinder was improved by adapting the mesh. Thefinite-rise bow shock and maximum overpressurewere more accurately predicted. The sampling dis-tance was increased for the adapted grid solution,which is important because typical aircraft requirea sampling distance of approximately one semispanto capture three-dimensional effects.

The solutions for the ONERA M6 wing withgrid refinement based on pressure gradient more ac-curately captured the lambda shock on the uppersurface. The adapted meshes captured a forwardshock which was not seen in the solution with auniform mesh; the shocks were more sharply de-fined. Increased local dissipation was required todamp pre- and post-shock pressure fluctuations onthe adapted mesh cases.

3D.TAG was successfully used to more sharplyresolve the bow shock of an unstarted inlet on a su-personic transport. Results presented in this reportdemonstrate that 3D_TAG and AIRPLANE havebeen successfully coupled and applied to realisticflow problems.

REFERENCES

Biswas, R., and Strawn, R. C., "A New Pro-cedure for Dynamic Adaption of Three-DimensionalUnstructured Grids," Applied Numerical Mathemat-ics, Vol. 13, No. 6, 1994, pp. 437-452.

2Jameson, A., Baker, T. J., and Weather-ill, N. P., "Calculation of Inviscid Transonic FlowOver a Complete Aircraft," AIAA Paper 86-0103,

AIAA 24th Aerospace Sciences Meeting, Reno, NV,Jan.1986.

3Jameson, A., and Baker, T. J., "Improvementsto the Aircraft Euler Method," AIAA Paper 87-0452, AIAA 25th Aerospace Sciences Meeting, Reno,NV, Jan. 1987.

4Barth, T. J.,"A 3-D Upwind Euler Solver forUnstructured Meshes," AIAA Paper 91-1548, AIAA10th Computational Fluid Dynamics Conference,Honolulu, HI, Jun. 1991.

5Strawn, R. C., Biswas, R., and Garceau, M.,"Unstructured Adaptive Mesh Computations of Ro-torcraft High-Speed Impulsive Noise," Journal ofAircraft, Vol. 32, No. 4, 1995, pp. 754-760.

6Duque, E. P. N., Biswas, R., and Strawn, R.C., "A Solution Adaptive Structured/UnstructuredOverset Grid Flow Solver with Applications to He-licopter Rotor Flows," AIAA Paper 95-1766, AIAA13th Applied Aerodynamics Conference, San Diego,CA, Jun. 1995.

7Baker, T. J., "Generation of TetrahedralMeshes Around Complete Aircraft," 2nd Interna-tional Conference on Numerical Grid Generation inComputational Fluid Dynamics, NASA and AirforceOffice of Scientific Research, Dec. 1988.

8Baker, T. J., "Automatic Mesh Generation forComplex Three-Dimensional Regions Using a Con-strained Delaunay Triangulation," Engineering withComputers, Vol. 5, Nos. 3 and 4, 1989, pp. 161-175.

9Jameson, A., Cliff, S. E., Thomas, S. D.,Baker, T. J., and Cheng, W-S., "Supersonic Trans-port Design on the IBM Parallel System SP2,"NASA Computational Aerosciences Workshop, Mof-fett Field, CA, Mar. 1995.

10Biswas, R., and Strawn, R. C., "Mesh Qual-ity Control for Multiply-Refined Tetrahedral Grids,"Applied Numerical Mathematics, to appear.

nThomas, C. L., "Extrapolation of Sonic BoomPressure Signatures by the Waveform ParameterMethod," NASA TN D-6832, Jun. 1972.

12Cliff, S. E., and Thomas, S. D., "Eu-ler/Experiment Correlations of Sonic Boom Pres-sure Signatures," Journal of Aircraft, Vol. 30, No. 5,1993, pp. 669-675.

13Schmitt, V., and Charpin, F., "Pressure Dis-tributions on the ONERA-M6-Wing at TransonicMach Numbers," AGARD Report AR-138, 1979.

14Melton, J. E., Thomas, S. D., and Cappuc-cio, G., "Unstructured Euler Flow Solutions usingHexahedral Cell Refinement," AIAA Paper 91-0637,AIAA 29th Aerospace Sciences Meeting, Reno, NV,Jan. 1991.

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