[American Institute of Aeronautics and Astronautics 39th Aerospace Sciences Meeting and Exhibit -...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-16266

AIAA 2001-0369Thermal Analysis of a High-SpeedAircraft Wing Using p-Version FiniteElementsDana C. GouldNASA LangleyHampton, VA

39th AIAA Aerospace SciencesMeeting & Exhibit

8-11 January 2001 / Reno, NV

For permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics,1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.


AIAA 2001-0369

THERMAL ANALYSIS OF A HIGH-SPEED AIRCRAFT WING USING p-VERSIONFINITE ELEMENTS

Dana C. Gould

NASA Langley Research CenterHampton, Virginia 23681-2199

ABSTRACTThis paper presents the results of conceptual levelthermal analyses of a High Speed Civil Transport(HSCT) wing using/7-version finite elements. The workwas motivated by a thermal analysis of a HSCT wingstructure which showed the importance of radiationheat transfer throughout the structure. The analysis alsoshowed that refining a traditional finite element mesh toaccurately capture the temperature distribution on theinternal structure led to very large meshes withunacceptably long execution times. Further studyindicated using p-version finite elements might improvecomputation performance for this class of problem.Methods for determining internal radiation heat transferwere then developed and demonstrated on test problemsrepresentative of the geometry found in an aircraft wingstructure.

This paper presents the results of the application ofthese new methods to the analysis of a high speedaircraft wing. Results for both a wing box model aswell as a full wing model are presented. The reducedwing box model allows for a comparison of thetraditional finite element method with mesh refinement(/z-refinement) to the newp-version finite elementswhile the full wing model demonstrates theapplicability and efficiency of p-version finite elementsfor large models.

INTRODUCTIONThis effort began with a study of the thermal analysis ofa high-speed civilian transport vehicle wing todetermine the current state of thermal analysiscapabilities for such a problem. That study, which usedcommercially available thermal analysis software, is

documented in an earlier paper.1 The primary difficultyin the analysis was accurately modeling the radiationheat transfer internal to the wing structure. Initialanalyses using a coarse mesh and including conductionand radiation heat transfer were run successfully;however, the large element size produced inaccuratetemperatures for the internal structure. To improve theresults, the analysis was repeated using a refined mesh.The initial run of the refined model considered heattransfer throughout the wing by conduction only, andproduced a solution with reasonable temperaturedistributions on the internal structure. Internal radiationexchange was then added to the model, but the solutionof the model never successfully completed.

To get temperature predictions for the internal structure,including the effects of internal radiation heat transfer,a reduced model was generated. The reduced modelincluded a single wing-box section of the wing. Thismodel was run both with and without internal radiationexchange, and the results showed that internal radiationexchange had a significant effect on the temperatures ofthe internal structure.

Thus a method that could analyze a full wing withinternal radiation was desired, and the application ofthep-method appeared promising based on the work inreference 1. This paper presents the reults of applyingthe methods developed in reference 1 to the thermalanalysis of a High Speed Civil Transport (HSCT) wing.

WING-BOX MODELBefore analyzing the full wing, a model of a singlewing-box was developed. The term wing-box refers tothe open area enclosed by adjacent ribs and spars andthe corresponding upper and lower surface sections of awing. The ribs and spars make up the internal structure

Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17,U.S. Code. The U.S. Government has a royalty^free license to exercise all rights under the copyright claimed herein for Governmental purposes.All other rights are reserved by the copyright owner.

1

American Institute of Aeronautics and Astronautics


of the wing with the spars running from the fuselage tothe wing tip, and the ribs running from the leading edgeof the wing to the trailing edge of the wing. The ribsand spars of the full wing model are shown in Figure 1.The wing-box model was developed so that results fromthe newp-element methods could be compared with theresults obtained from traditional methods.

Typicalrib

t —Starboard rib

typical spar

Figure 1: High Speed Civil Transport wing geometryincluding skin, ribs, and spars.

The wing-box, shown in Figure 2, is 60 inches long(fore to aft) by 15 inches wide (port to starboard) by 30inches high (thickness of wing). The vertical surfacesof the wing-box represent the internal structure (ribsand spars) and are adiabatic on their external faces.The upper and lower surfaces of the wing-box representthe upper and lower surfaces of the wing skin and aresubjected to convective heating due to the airflow overthe wing. The convection boundary conditions on theupper and lower surface were taken from a mid-winglocation of the full wing model described in thefollowing sections. The methods used to generate theconvection boundary conditions are discussed in thefollowing section as well. Convective heating wasapplied to the upper and lower surfaces of the wing-boxassuming uniform flow over the surfaces. Theconvection coefficients for both the upper and lowersurfaces are given in Table 1 and are assumed to remainconstant throughout the trajectory. Since there is littledifference in the recovery temperatures for the upperand lower surfaces, the same time-dependent valueswere used for both and are listed in Table 2. Heattransfer due to natural convection internal to the wing-box was ignored.

Convective heatingon upper surface

Convective heatingon lower surface

Dimensions in inches

Figure 2: Geometry of wing-box model.

Table 1: Convection coefficients for wing-box model.

Surface

Upper

Lower

Convection Coefficient

BTU/hr-in2-°F

0.06

0.84

Table 2: Convection recovery temperatures for wing-box model.

Time

(hours)

0.00

0.20

4.37

5.00

Fluid Temperature

(°F)

59

334

336

61

The six surfaces forming the wing-box use acorrugated panel construction as shown in Figure 3.The panels are made from titanium and the thermalproperties for this material are listed in Table 3. Thecorrugated panels are modeled with isotropic elementswith an equivalent thickness of 0.065 inches. Thisthickness represents the thickness of the face sheet(0.015 inches) plus the thickness of the core sheet (0.05inches). All of the internal faces of the wing-boxradiate to each other as diffuse-gray surfaces, and theupper and lower skin surfaces also radiate to space. All



surfaces have an emissivity of 0.85, and the uppersurface has an absorbed heat flux of 1.5 BTU/hr-in2 dueto solar heating. The lower surface has a combinedsolar and earth IR heating of 0.89 BTU/hr-in2.

Figure 3: Corrugated panel construction.

Table 3: Thermal properties used in wing-box model.

Material

Density

Specific heat

thermal conductivity

Titanium

0.16(lb/in3)

0.135(BTU/lb-°F)

0.343(BTU/hr-inch-°F)

Analysis with ̂ -refinementThe wing-box was analyzed using the new /7-elementmethods with a single element for each of the sixsurfaces of the wing-box. Cases were run with;? valuesranging from 1 to 6. Results were obtained for both theradiation sub element and integration methods (seereference 1). Each analysis was run using a time step of0.1, 0.05, and 0.01 hours with temperature resultsoutput at every time step. Results for the different timesteps were similar and the results presented here arefrom the runs using a 0.05 hour time step. Figure 4shows temperature contours at time = 0.3 hours for thecase of p=l using the radiation sub-element method andFigure 5 shows the same case with p=2 elements. Forp=l, the radiation sub-element approach reverts to thetraditional methods similar ro those employed incommercial finite element heat transfer software.Comparing the contours for the p=l case to the p=2case, significant differences can be seen because thep=l mesh does not adequately model the side walls ofthe wing-box. For the p=l case, the nodes for the sidewall elements are located on upper and lower surfaceswith no nodes in-between. With no nodes between theupper and lower surfaces, the temperature of thesidewalls cannot be computed accurately. The nodeson the upper and lower surfaces are shared by the upperand lower surface elements which have the convection

boundary conditions. The convection boundaryconditions drive the temperatures of these nodes and thetemperature lag midway between the upper and lowersurfaces is not captured.

Figure 6 shows temperature contours for the integrationmethod case with p= 4 elements, at 0.3 hours aftertakeoff. The results are similar to the p=2 results inFigure 5, with the temperatures in the middle of thevertical surfaces (the internal structure) slightlywarmer. In fact, the results do not change significantlyfor higher-order elements. Figure 7 shows the resultsfor the integration method using p=6 elements. In thecases run here, the results for the radiation sub elementmethod and the integration method were similar.

To see the transient nature of the temperatures gradientsin the wing-box, two temperatures from the forwardrib-port spar junction have been plotted versus time.Point A is at the intersection of the fore spar, the portrib, and the lower surface while point B is along thespar-rib edge halfway between the upper and lowersurfaces (see Figure 7). Figure 8 shows the temperaturehistories computed using the radiation sub-elementmethod with p=2 elements, and Figure 9 shows theresults from the integration method using the p=6results. Figure 8 shows that point B initially drops intemperature with the p=2 elements. This behavior is aconsequence of using a consistent mass matrix in thetransient solution algorithm in a problem with rapidtemperature change (see, for example, p. 334 of [2]).The temperature drop had the same approximatemagnitude and duration regardless of the time step used(0.1 hours, 0.05 hours, 0.01 hours). Switching to abackwards difference for the time derivative eliminatedthe temperature drop; however, the backwardsdifference is only first-order accurate whereas thecentral difference used in the Crank-Nicholson methodis second-order accurate. Note however, that thetemperature drop did not occur for the higher-orderelements as shown Figure 9 for the p=6 elements. Loboand Emery [3] studied this behavior and concluded thatthe higher-order elements are not immune from thisanomalous behavior but simply less susceptible to it.


(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

Figure 4: Temperature contours of the wing-box modelusing the radiation sub element method with p=lelements at 0.3 hours.

Figure 6: Temperature contours of the wing-box modelusing the integration method with p=4 elements at 0.3hours.

Figure 5: Temperature contours of the wing-box modelusing the radiation sub-element method with p=2elements at 0.3 hours.

Figure 7: Temperature contours of the wing-box modelusing the integration method with p=6 elements at 0.3hours.



300

250

200

100

2 3time [hours]

200

150

100

-50

Figure 8: Transient temperature response at selectedpoints of the wing-box model using the radiation sub-element method with p=2 elements. (Location of pointsshown in Figure 7.)

The traditional method does however have oneadvantage, speed. Table 4 lists the number of degreesof freedom and the execution time for several cases.The execution times were obtained on a 133 MHzPentium PC with 32 Mbytes of RAM. For a givennumber of degrees of freedom the traditional method isfaster than the radiation sub-element method which inturn is faster than the integration method. Thedifferences in run times increase significantly as theelement order increases in the ̂ -method. However, acomparison based on the degrees of freedom does notaccount for the improved accuracy of the higher-ordermethods. The contour plots show the p=2 solution tobe the most similar to the traditional method with a 10by 10 mesh (the most refined mesh used) but with asignificantly lower execution time. This result isconsistent with the results in reference [1] where a morerigorous accuracy comparison of the methods waspresented.

- 200

Figure 9: Transient temperature response at selectedpoints of the wing-box model using the integrationmethod with p=6 elements. (Location of points shownin Figure 7.)

Analysis with /z-refinementSeveral finite element analyses were made of the wing-box using the traditional radiation methods with h-refinement. In these analyses, each surface wassubdivided into an n by n mesh where n varied from 1to 10 (the view factor code was the limiting factor inincreasing the mesh size). Results for the 2 by 2, 5 by5, and 10 by 10 meshes are shown in Figure 10-12. The/z-reiinement behavior shown in these figures is similarto the behavior seen in the /7-enrichment cases, but notethat the lowest order/? results (p=2) are similar to theresults from the largest n by n mesh case.

Figure 10: Temperature contours for wing-box modelcase 12 (2 elements per edge or 4 elements per surface)at 0.3 hours.



Figure 11: Temperature contours for wing-box modelcase 14 (5 elements per edge, 25 elements per surface)at 0.3 hours.

Figure 12: Temperature contours for wing-box modelcase 15(10 elements per edge, 100 elements persurface) at 0.3 hours.

Table 4: Execution times for wing-box model usingnew methods versus traditional methods.

Case

123456789101112131415

Method*

RSERSERSERSERSEIMIMIMIMIMTMTMTMTMTM

elementP234562345611111

elementsper edge

(n)1111111111123510

#ofDOF265698152218265698152218

82656152602

Execu-tion time

(sec.)72

L 193502126127333159962867677716200

9.13784

36017018

*RSE = Radiation sub-element method, IM =Integration method, TM = Traditional Method

FULL WING ANALYSISDescriptionThe next analysis is of a complete HSCT wing. Theanalysis presented here is typical of a conceptual leveldesign effort that integrates several disciplines toevaluate the feasibility of a proposed vehicle. Thesystem analysis begins with an aerodynamic analysis ofthe proposed vehicle at selected points throughout thetrajectory. Aerodynamic heating data is then used in athermal analysis of the vehicle to predict temperaturesthroughout the structure. These temperatures are thentransferred to a structural model of the vehicle alongwith the pressure loads from the aerodynamic analysis.The structural model is used to size the load-carryingcomponents of the vehicle, typically through anoptimization process. The final component sizingdetermines component level and overall vehicle weight.The weight data, trajectory data, and aerodynamic dataare then used in a trajectory analysis to determine theoverall viability of the vehicle. The entire process isthen iterated until a final optimized design isdetermined.

The conceptual level design used in this processproduces thermal and structural models with simplegeometric elements. For example, holes thatcommonly occur in the internal structure of the wingare not considered at this stage, but are left to thedetailed design process. The goal of the thermal andstructural analyses in this process is not to analyze a

6



particular design in great detail, but rather to accuratelyevaluate multiple conceptual designs in order to arriveat an optimum structure.

The thermal analysis of an entire vehicle is oftenperformed using individual models for the majorcomponents of the vehicle. The analysis presented hereis of a wing structure of a high-speed civil transport.The geometry used in the analysis was supplied by theGEOLAB group at NASA Langley Research Center.The wing, shown in Figure 1, is approximately 113 feetlong at the root, and 55 feet wide at the trailing edge.The structure consists of the upper and lower skinsurfaces of the wing as well as the internal structure ofribs and spars. The skin surfaces represent the outermold line of the wing and are consistent with thegeometry used for aerodynamic analysis.

The wing skin uses the same corrugated panelconstruction used in the wing-box model, which isshown in Figure 3. The panels were modeled using thesame material properties used for the wing-box modelwhich are listed in Table 3. As in the wing-box model,the corrugated panels are modeled with isotropicelements with an equivalent thickness of 0.065 inches.This thickness represents the thickness of the face sheet(0.015 inches) plus the thickness of the core sheet (0.05inches). For simplicity, the internal structure wasassumed to be made of the same corrugated panelconstruction.

Aerodynamic HeatingA five-hour flight trajectory representative of acommercial airline or transport route was used in theanalysis. The Mach 2.4 cruise lasts 3.8 hours andreaches a maximum altitude of 70,000 feet. Figure 13shows the altitude and speed of the vehicle throughoutthe trajectory, the analysis does not include any specialmaneuvers (for example an emergency descent) whichoften drive the design of an aircraft.

Altitude(kft)

MachNumber

Figure 13: Mach number and altitude for commercialroute of a High Speed Civil Transport.

To compute the convection boundary conditions overthe full wing, NASA Langley's version of theMINIVER computer code, LANMIN [4], was used.The upper and lower surfaces were subdivided to giveone surface element between adjacent pairs of ribs andspars. The surface elements of Figure 1 represent thesurface elements used to compute the convectionboundary conditions. The centroid of each surface wascomputed along with the distance from the leading edgeof the wing to the surface centroid. This data was thencombined with the trajectory data to compute aconvection coefficient and recovery temperature forevery surface element in Figure 1 at every trajectorypoint. Upon inspection of the data, and based on resultsof some wing-box model runs, it was determined thatthe convection boundary condition could be accuratelymodeled using only four trajectory points andinterpolating the data between points. This greatlyreduced the amount of data that had to be transferred tothe finite element model without a significant loss inaccuracy.

Full Wing Finite Element Thermal ModelThe full wing model was meshed using one element torepresent each surface shown in Figure 1. Thus themesh used in the aerodynamic heating analysis iscoincident with the thermal analysis mesh and eachwing surface element has a separate transientconvection boundary condition associated with it. Thewing was assumed to be totally isolated from thefuselage in this model, thus an adiabatic boundarycondition was assumed at the fuselage. The internalsurfaces of the wing were assumed to be diffuse-graysurfaces with an emissivity of 0.85. Results from thewing-box model showed that the convection boundarycondition dominated the surface heating conditionssuch that external radiation and solar and earth IR heatfluxes had negligible effect, so they were not includedin the full wing analysis. Natural convection inside thewing was also not considered.

Due to the large number of elements, and the problemsencountered with view factor computations for suchlarge models in the initial study, only the integrationmethod was used to analyze the full wing model. Themodel consists of 235 elements and 725 nodes, and wasanalyzed using p=2, p=3, and p=4 elements. There are725 degrees of freedom in the model using p=2elements, 1769 degrees of freedom in the p=3 elementmodel, and 3283 degrees of freedom in the p=4 elementmodel. The model with p=2 elements was run withtime steps of 0.1, 0.05, and 0.01 hours with nosignificant differences in the results. The p=3 and p=4runs used a time step of 0.05 hours. All runs were



performed on a 200 MHz Pentium class personalcomputer.

A temperature contour plot of the upper surface of thewing at the beginning of the cruise portion of thetrajectory (0.2 hours into the flight) is shown in Figure14. First note that the temperature variation over theentire wing surface is not that large, 10 to 15 °F.Except for the outboard leading edge, the lowesttemperatures on the wing occur near the fuselage wherethe wing is thickest. The deeper ribs and spars in thisarea act as heat sinks preventing the surroundingstructure from heating as fast as other areas. Movingout from the fuselage (+y) the wing thins and theresulting internal structure becomes smaller. Thesmaller internal structure provides less thermal massand the corresponding skin temperatures increase. Nearthe fuselage, the heat sink effect of the relativelymassive internal structure can be clearly seen by theoval contours corresponding to individual wing-boxes.The oval contours occur because the temperature at theedges of the wing-boxes are held down by the adjacentinternal structure while the temperatures at the center ofthe wing-box respond more rapidly. Figure 15 showsthe corresponding temperature contours for the lowersurface. Since the lower surface has a higherconvection coefficient, it heats up faster than the uppersurface. This is evidenced by the overall highertemperature in Figure 15 as compared to Figure 14.The temperature gradients on both surfaces disappearquickly, and are in fact gone by 0.3 hours as shown inFigure 16 and Figure 17.

To see the temperature response of the internalstructure, the data for a spar, a rib, and a single wing-box have been extracted from the full wing data set.The approximate locations of the components areshown in Figure 17. Figure 18 shows the temperaturecontours for the selected rib at 0.2 hours. The samegeneral patterns that were seen in the wing-box modelappear here. The upper and lower surfaces of the sparare at uniform temperatures while the temperaturesbetween the upper and lower surfaces are cooler. Thelowest temperatures occur where the spars intersects therib (vertical lines in Figure 18) halfway between theupper and lower surfaces. These patterns are alsoshown in Figure 19 where the temperatures along theupper and lower surface of the spar are plotted alongwith the temperature halfway between the upper andlower surfaces. The symbols on the line plot are usedonly to differentiate the curves, they do not representthe actual data points (each curve was generated usingapproximately 60 equally spaced temperature values).The mid-plane temperature profile shows a steep dropat the intersection of each spar. Figure 20 shows the

same temperature plots at 0.3 hours. The temperaturevariation along the mid-plane at this later time showsthe same patterns that were present at 0.2 hours,although the overall temperature is higher and themagnitude of the variations has diminished. Figure 21shows the results from the model with p=4 elements at0.2 hours. The results are similar to the p=2 results withthe p=4 elements producing a flatter temperature profilenear the center of the wing-boxes.

The temperature contours for the selected spar at 0.2hours using p=2 elements are shown in Figure 22.Once again the upper and lower surfaces are relativelywarm with uniform temperatures. The lowesttemperatures occur midway between the upper andlower surfaces where the ribs intersect the spar. As thewing thins moving out from the fuselage, the mid-planetemperatures (halfway between the upper and lowersurfaces) increase and the gradients in the spardimmish. This pattern is also illustrated in Figure 23where the temperatures along the upper, lower, andmid-plane surfaces for this spar at 0.2 hours are plotted.Figure 24 shows the same temperature plots at 0.3hours. Once again, the overall structure temperaturehas increased and the magnitude of the temperaturegradients have decreased relative to the earlier timepoint. Figure 25 shows the results from the model withp=4 elements at 0.2 hours. Again the results are similarto the p=2 results with the p=4 elements producing aflatter temperature profile near the center of the wing-boxes.

Finally, Figure 26 shows the temperature contours ofthe selected wing-box using the p=2 results at 0.3hours. This wing-box is one of the larger wing-boxesin the wing and thus has some of the largesttemperature gradients. The contours shown in Figure26 have the same patterns as seen in the wing-boxanalysis. The coolest temperatures occur at theintersection of the ribs and spars halfway between theupper and lower surface. One slight difference is thatthe outboard rib is shorter than the inboard rib andresults in lower gradients across the outboard section ofthe wing-box.

8



rT[°F]

I 11 33510 330

Figure 14: Temperature contours of the full wing modelupper surface at 0.2 hours using p=2 elements.

Figure 17: Temperature contours of the full wing modellower surface at 0.3 hours using p=2 elements.

L,

Figure 15:Temperature contours of the full wing modellower surface at 0.2 hours using p=2 elements.

Note: z dimension scale expanded to 5 times x dimension scale

Figure 18: Temperature contour for selected rib of fullwing model at 0.2 hours using p=2 elements.

Figure 16: Temperature contours of the full wing modelupper surface at 0.3 hours using p=2 elements.

1800 2000 2200x [inches] 2400

Figure 19: Temperatures along selected rib of full wingmodel at 0.2 hours using p=2 elements.



340

320

300

280

E260H 240

220

200

180

160

2000 2200x [inches]

2400


Figure 23: Temperatures along selected spar of fullwing model at 0.2 hours using p=2 elements.

1800 2000 2200x [inches] 2400


320

300

280

IT 260£_,H 240

220

200

180

160

outboard direction

600 400y [inches]

200


Note: z dimension scale expanded to 5 times x dimension scale

Figure 22: Temperature contour for selected spar of fullwing model at 0.2 hours using p=2 elements.

10



200 fuselage edge


Figure 26: Typical wing-box extracted from full wingmodel, time = 0.3 hours, p=2 elements.

CONCLUSIONSThe methods developed in [1] have been used

for the thermal analysis of a High Speed CivilTransport wing using a wing-box model as well as afull wing model. Results from the full wing modelindicate that a solution with p=2 elements provides agood balance between accuracy and cost (executiontime). The development of these methods was theresult of an initial study that demonstrated thedifficulties in analyzing a full wing model due to thecomplexities involved with the internal radiation heatexchange. The new methods provide a way to analyzethe full wing structure accurately and efficiently. Thiscapability is particularly useful in the conceptual design

stage of a vehicle when overall design parameters are ofinterest. The new methods will permit a better heattransfer representation of the internal structure and willthus give a more accurate prediction of the temperaturedistribution throughout the wing. The improvedtemperature distribution can then be used in a structuralmodel of the wing to evaluate thermal stress. Therefined temperature distribution will improve thethermal stress distribution producing a more accuratesizing of the structural components of the wing, animportant factor since these vehicles are extremelyweight critical.

Gould, D. C., " Radiation Heat TransferBetween Diffuse-Gray Surfaces Using Higher-Order Finite Elements", AIAA Paper 2000-2371,2000.

Huebner, K. H., Thornton, E. A., Byrom, T. G.,The Finite Element Method For Engineers, 3rd

edition, John Wiley and Sons, New York, 1995.

Lobo, M., Emery, A. F., "The DiscreteMaximum Principle I Finite-Element ThermalRadiation Analysis", Numerical Heat Transfer,Part B, Vol. 24, pp. 209-227, 1993.

Engel, C. D., Praharaj, S. C., MINIVER Upgradefor The A VID System, Volume 1: LANMINUsers's Manual, NASA CR-172212, 1983.

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