Numerical Analysis - Ablation

7/31/2019 Numerical Analysis - Ablation

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NUMERICAL ANALYSIS OFTHE TRANSIENT RESPONSE OF ABLATINGAXISYMMETRICBODIES INCLUDINGTHE EFFECTS OF SHAPE CHANGEby Stephen S. Tompkins, James N. Moss,Claud M . Pittmun, and Lonu M . HowserLangiey Research CenterHdmpton, Va. 23365N A T I O N A L A E R O N A U T I C S A N D S P A C E ADMINISTRATION W A S H I N G T O N , D C. M AY 1 9 7 1


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~

1.ReportNo. 2. Government Accession No.NASA TN D-6220

4. Title and SubtitleNUMERICAL ANALYSIS OF THE TRANSIENT RESPONSE OF IABLATING AXISYMMETRIC BODIES INCLUDING THEEFF ECTS OF SHAPE CHANGE IStephen S. Tompk ins , Jam es N. Moss, Claud M. Pittman, andLona M. Howser

7. Author(s) 1[9. Performing Organization Name and AddressNASA Langley Research CenterHampton, Va. 23365

12. SponsoringAgencyNameandAddressNational Aeronautics and Space Administration -

~ Washington, D.C. 20546~ 15. SupplementaryNotes

3. Recipient's Catalog No.

5. Report DateMay 19716. PerformingOrganization Code

8. Performing Organization Report No.L-747410. Work Unit No.

124-07-26-0211. Contract or Grant No.

13. Type of Report andPeriodCoveredTechnical Note

14. SponsoringAgency Code

16. Abstract

The differential equations governing the transient response of an ablating axisym-metric, orthotropic body have been derived for fixed points in a moving coordinate system.These equations have been e xpanded into finite-difference form and programed for numer-ic al solution, with an implicit technique, on a digital computer. Numerical results comparefavorably with exact solutions.

Several applications of the analysis are discussed. These applications demonstratethe sig nificance of a salie nt featu re of the analysis, that is , the ability to analyze the effectsof changes in body ge omet ry. This feat ure was used to obtain satisfactory agreementbetween numerical and experimental results for an ablating teflon sphere and a small testspecimen exposed to a high-intensity laser beam. Although the anal ysis is primarily for asingle-layer material, a multilayer material can be successfully approximated under cer-tain conditions by a single-layer system.

17.Key Words Suggested by Author is) )AblationHeat transferNumerical analysis

18. Distribution StatementUnclassified - Unlimited

I19. Security Classif. (of this report) 22. Price.1. No. ofPages0. Security Classif. (of this page)

Unclassified $3.003nclassified- . - .For sele by he National Technical Information ervice, Springfield, Virginia 22151


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NUMERICAL ANALYSIS OF THE TRANSIENT RESPONSE OFABLATING AXISYMMETRIC BODIES INCLUDING

THE EFFECTS OF SHAPE CHANGEBy Stephen S. Tompkins, James N. Moss, Claud M. Pittman,

and Lona M. HowserLangley Research Center

SUMMARYThe differential equations governing the transient response of an ablating axisym-

met ric, orth otropic body have been derived for fixed points ina moving coordinate sys-tem. These equations have been expanded into finite-difference form and programed fornumerical solution, with an implic it technique, on a digital computer. Numerical resultscompare favorably with exac t solutions.

Several applications of the analysis are discussed. These applications demonstra tethe significance of a sali ent feat ure of the analysis, that is, the ability to analyze theeffects of changes in body geomet ry. This featu re w as used to obtain satisfactory agree-ment between numerical and experimental results for an ablating teflon sphere andasmall test specimen exposed to a high-intensity laser beam. Although the analys is isprimarily for a single-layer material, a multilayer material can be successfully approx-imated under certain conditions by a single-layer system.

INTRODUCTIONOne-dimensional ablation analyses have been used extensively to study the thermal

resp onse of heat shields subjected to aerodynamic heating. However, for heat shieldswith large curvatu re or large heating-rate variations over the surface, the assumptionsof one-dimensional heat flow no longer apply, and an accurate descript ion of the thermalresponse requires an ablation analysis for multidimensional heat transfer that includesthe effects of changesn eat-shield eometry.

References 1 o 6 are exam ple s of two-dimensional thermal analyses presentlyavailab le. The outstanding features of the analyses of refe ren ces 1 and 2 ar e tha t bothconsider axisymmetric bodies of an isotrop ic material s and both use stable, implicitnumerical methods to solve the heat-conduction equation. However, neither analysisconsiders mass transfer. Mass t ransfer is included in the analysis of re fer ence 3 in


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addition to the indicated features of ref ere nces 1 and 2. The analysis of reference 3 isformulated for fixed nodal points in fixed co ordinate system a nd ofte n requir es inter-polation at the boundaries because the nodal points and the boundaries do not alwayscoincide. This interpolation can lead to inaccuracies. The analysis of refe ren ce 4 issimilar to th at of r efer ence 3 except that it uses a moving coordinate system (whicheliminates interpolation) and a conditionally stable, time-consuming explicit formulationas opposed to the stable, time-saving implicit formulation used in references 1 o 3 .Only references 5 and 6 conside r the effec ts of shape change on the thermal response ofthe heat shield. However, reference 5 does not consider an anisotropic material, andboth references 5 and 6 use explicit methods to solve the governing equations.

Collectively, references 1 to 6 consider many of the signif icant physical character-ist ics of an axisymmetric ablating heat shield and also demonstrate the desirable methodof solution, that is , an implicit method. However, no sing le refe renc e inco rpor ates allthese features into one analysis.

This paper presents and discusses a transient two-dimensional ablation analysiswhich incorporates all the significant characteristics of the physical problem consideredcollectively in references 1 to 6. The analysis has the following feature s: (1)the abla-tion material is considered to be orthotropic with temperature-dependent thermal prop-erties; (2) the t herm al re spon se of the e ntir e body is considered simuitaneously; (3 ) theheat-transfer and pressure distribution over the body are adjusted to the new geometryas ablation occurs; (4 ) the governing equations and several boundary-condition optionsa r e formula ted in term s of generalized orthogonal coordinates for fixed points inamoving coordinate system; and (5) the finite-difference equations are der ive d and solvedimplicitly.

The accuracy of the analysis presented in this paper is demonstrated by compari-sons between numerical results and exact solutions for simplified conduction problems.Selected examples and test da ta are shown to demonstr ate the u tility of the analysis andthe importance of the coupling between body-shape change and the heat ing- rate and pres-sure distributions.

The finite-difference equations have been programed for solution on a high-speeddigital computer. The program has a plotting routine which can display the shape of theablating body at any timeuringhealculation.

SYMBOLS1 a 6A = - -Xb defined by equation (10)

AC constantnxidationquationorrespondingopecificeaction rate2


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coeffic ients in equation (28)

AS constantnublimationquationa constanto djust quations (C24),C25),nd (C26) to orr ect orm or

Cartesian coordinates

BC constantnxponential of oxidationquationorrespondingoctivationenergy

BS constantnxponential of sublimation equationC oxygen concentration by mass

cP specific heatH totalnthalpy

AHC heat of combustion

AHS heat of sublimationh17h2,h3 oordinate cale actors eqs. (2))i order of reaction (eq. (11))K reaction-rateonstantorxidationeq. (15))kL number of stationsn-directionM moleculareight of gasM molecular weight of oxygen0 2


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mo,mSPPW

qCq c net%et

q rR

RcylRstag

integersmass loss ra terate at which oxygen diffuses to surfacemass loss due to sublimationexponent of p ressure in subl imation equatio n (eq. (17))wall pressureconvective heating r a te to nonablating cold wa l l

hot-wall convective heating rate corre cted for trans pirati on (eq. (13))net heating rate to surface including combustion, sublimation,and surface

reradiation (eq. (19))radiant heating ratera dius of curva tur e of base curv ecylindrical radius from axis of symmetry to base curvestagnation-point radius of curvature

r exponent of radiusnublimationquation (17); phericaloordinateS number of stat ionsn-directionT

TBTm,nemperature at finite-differencetation (m,n)

temp erature of body to which back surface radiates

t thickness of heatin k

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X,Y

P6

7

cp

free-stream velocityCartesian coordinates (see fig. 2)curvilinear coordinates (see fig. 1)length of base curveabsorptanceweighting factor for transpiration effectivenessof mass loss due to

combustionweighting factor fo r transp iration e ffectiveness of m as s lo ss due to

sublimationeither 0 or 1 depending on whether tran spir atio n o r ablation theory is usedmaterial thicknessemittanceanglebetween R andRcyl (fig. 1);sphericalcoordinate

ma ss of c har remo ved per uni t ma ss of oxygendimensionless curvilinear coordinates, equations (4 )density of materialStefan-Boltzmann constanttimeangle of rotation about axis of symmetry, figure 3angle between axis of symmetry and normal to surface, figure1

5


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Subscripts:

C combustione edge of boundaryayermn integersL last stationn-directionmax0 original,alue at previousime

S last stationn-directionstagtagnation-pointonditionW w a l l conditionX,Y coordinates5,rl dimensionlessoordinatesSuperscripts:I conditionlong x = LI 1 conditionlong = 0

ANALYSISPhysical Model

The analysis con siders an axisym metric ablating body exposed to aerodynamicheating; this body is composed of a single orthotropic materialof varying thickness withtemperature- dependent thermal properties. (Seefig. 1.) Although the ana lys is con-s iders a single-layer material, the analysis of a multilayer material, such as a charringablator, c an be suc cessfully a pproximated with the present an alysis und er certain ondi-tions. (See ref. 7, forexample.)6


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Surface normal

Or tho t rop i c mater i a l

k > P> k(X,Y)]ULV Body axis of symmetry-igure 1.- Schem atic diagram of the physical model for a typical axidsymmetric body.Two coordinate systems are used to study the thermal and ablative responseof the

heat shield. One is a curvilinear coordinate system with x,y coordinates (fig. l ) ,whichis used to determine internal temperature distributions. A stationary base curve locatedat the back surface of the ablator es tablishes the x-axis.

The second coordinate system (fig. 2 ) is used to define the heat-shield exterior

"1. I n i t i a l s u r f a c ec urface a t t ime r

Figure 2. - Coordinate system used o define body geometry.

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geometry which changes with time as a res ult of ablation. This coordina te system, withw ,z coordinates, is a Cartesian system with the origin fixedat the original stagnationpoint on the heat shield. All the geometric parameters needed to compute changes in thestagnation heating rates and the heating-rate and pressure distributions over the surfaceare defined in this system. The equations that quantitatively relate the internal temper-atur es with the changes in the heating and pressure over the surface f the heat shieldare given in the following sec tions.

Governing Differential EquationsThe differential a r c length ds in the curvilinear coordinate system in figure 3 is

= h:(dx)2 + h 2 ( d ~ ) ~h3(dq)2where the scale factors are

h l = 1 + -R

The curvilinear coordinate system should conveniently describe any axisym metr icbody geometry of in te re st . However, if the Cartesian coordinate system is required,care m ust be taken to use unity scal e factors. Unity sc ale fac tor s a re obtained byassigning hevalues of 1 toRcyl, 00 to R , and n/2 to 6. (See qs. (2).)

For an axisy mme tri c body, the governing time-dependent heat-conduction equationwith variable coefficients is (in fixed coordinates)

c

If equation (3) were expressed in finite-difference form, it would describe the tem-perature variation at fixed stations in a fixed coordinate system. To maintain a fixednumber of stations in a layer which changes thickness with time, it is necessary tochange the locations of the stations and to interpola te to determine the temperatures atthe new location after each time step in the calculation. This procedure not onlyincreases the time required to perform the computationsut also introduces a smallerror in each stepof the calculation. An alternative to this procedure is to transform8


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Figure 3 . - Basiccurv i l inea rcoordinate system.the equation to a coordinate system inwhich the stat ions rema in fixed and the coord inatesthemselv es move to accommodate changes in the surface location.

This transformation canbe made by introducing a moving coordinate system whichis defined by the following relations:

Inat

this system, the outer surface remains fixed at q = 1 and all other stations remainfixed values of 7.

Before equation (3) can be transformed to the[ , q coordinates, derivatives withrespect o x andy in erms of 5 and q must bedetermined.Thesederivativesare given in the following equations:

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and

Because is a function of 6 which is also a function of time , he ime derivat ive onthe right side of equation (3) becomes

Thederivatives of f and q are, romequations (4),

A change in 6 is given by

Therefore,

Replacing the appropriate terms in equation (7a) with equations (7b) and (7d) gives

Substituting equations (2b), (5), (6), nd (8) into equation (3 ) gives, in the trans-form ed moving coordinates,

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where

The solution to equation (9) depends on the initial conditions and the specifiedboundary conditions. These conditions are dis cus sed in t he following sections.Initial Conditions

The initial conditions that must be specified are the temp erat ure distr ibut ion,mass-transfer rates, and the body shape. For mos t cas es of interes t, the initial tem-perature distributionis uniform and the initial mass-transfer rate is zero.

Surface Boundary ConditionsTwo conditions must be specified at the outer surface. Either the rate of removalof the surface material o r the surfa ce tempe rature mu st be s pecified; the othe r ondition

is provided by an energy balance at the surface.Surface recession.- Ablation is assumed to result from a chemical process (oxida-

tion) or from a phase-change process (sublimation). The rate of m ass loss due to oxida-tion by molecular oxygen is , for an ith-order reaction,

In this analysis, it is assumed that all oxygen at the surface is in molecular form.The net rate at which oxygen d iffuses to the surface is , from reference 8 (assuming

a unit Lewis number),

where

11


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which is the net convective heating ra te to a hot ablating surface. Either transpirationtheory ( p = 0) or linear ablation theory ( p = 1) can be used to account for the effects ofmass transfer on the convective heating rate.

The rate at which mass is removed by oxygen must be proportional to the net rateat which oxygen diffuses to the surface; that s,

mc = A m @

whereK = Ace -Bc/Tw

The equation for a first-order oxidation reaction (i = 1)is

Equations (15) and (16) apply to both the reaction-rate-controlled and the diffusion-controlled oxidation regimes as well as the intermediate conditions.

Th e ra te of ablation by sublim ation isAS(Pw)P -Bs/Tw(Rs tag)r

ms = e

The form of equation (17) is compatible with reference 9 and most other sublimationtheories. Either equation (15), (16), or (17) can be the boundary condition that definesthe rate of removal of the surface material.12


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Surface location.- The thickness at any point 5 at any tim e is

The mass -los s vari ation about the body caused by the heating-rate distribution willresult in a nonuniform change in the geometry of the outer surface. This effect will i nturn affect the heating and pressure distribution s discussed in a subsequent section.

Surface energy balance.- The heat input to the surface consists of convective andradiative heating and heat from combustion when oxidation occurs. This heat inpu t mustbe accommodated by one or more of the following mechanisms: (1)aerodynamic blockingby mass transfer, (2 ) reradiation from the surface, (3) conduction into the material, and(4 ) sublimation of the surface material.

The surface energy balance is

where

The heat absorbed during sublimation and the heat released during oxidation areconsidered separately in equation (20) as is the blocking effectiveness of the gases pro-duced by oxidation and sublimation in equation (13). In the present analysis, oxidationand sublimation are not allowed to occur simultaneously.

The mass transfer affects only convective heating in this analysis. Reference 10indicates that at hypersonic entry velocities, radiant heating may also be significantlyaffected by mass trans fer. Howe ver, at present there is no quantitative analysis for theeffect of m as s t ra ns fe r on radiant heating, and it is therefore neglected. Additionalterms can easily be included in equation (20) to account for other phenomena which mayaffect the energy input to the surface.

Cor rect ion for change in body geometry .- The heat input to the sur face s alsoaffected by changes in body geometry. In the presen t analysis, the stagnation convectiveand radiative heat-transfer rates are adjusted for changes in the body bluntness asfollows:

~~

&.tag,o/R,tag and qr = qr,o Rstag,oSC =9c,0=- %tag


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Geometry changes affect the heating and pressure distribution around the ody aswell. The heating-rate distribution is computed by using equation (14) of reference 11and modified Newtonian press ure distribut ion. The methods for evalua ting the bodyshape paramete rs which are used to determine the heating-rate and pressure distribu-tions are given in appendix A.Modified Newtonian pre ssur e dis trib utio ns are suff icie ntly a ccur ate fo r mostppli-cations where the body is a hemisphere or a hemispherically blunted cone. For cones,the cone angle must be less than the value required to maintain supersonic low over theconical portion of the body. Also, for ablating hemispheri cal o r hemispherically cappedbodies, the modified Newtonian pressure distribut ion and consequ ently the heating- ratedistributions become inaccurate as the body sh ap es ar e blunted; that is, as the ratio ofthe body radius to nose radi us approa ches zero .

Boundary Conditions Along the


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and

The boundary conditions along this line of symmetry at the surface 5 = 0, ?? = 1and at q = 0 are the same as those previously described. However, this line is mathe-matically a singularity for equation (3) . The equations required along this line are dis-cussed in the following section.

SingularitiesEquation (3) applies to t he e ntire regionof interest; however, a line of discontinu ity

(singularity) exists. An inspection of equation (2c) shows that at x = 0, that is, along theline of sym metr y, the scal e fact or h3 vanishes. This coordinate singularity can beeliminated by using proper approximations valid only near x = 0. Appendix B presentsa detailed deriva tion of a form of equation (3) that applies at x = 0 and is given in [,qcoordinates as

Theboundaryconditions at [ = 0 are, at q = 0,

and, at 17 = 1,

METHOD OF SOLUTIONThe differential equations that define the temperature field in an ablating axisym-

me tr ic body of revolution a r e given in the previous sec tion. To obtain a solution, theseequations have been approximated by finite-difference equations and programed for solu-tion on a high-speed digital computer. The methods used to derive the finite-difference

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expressions are given in appendix C along with a summary of the finite-difference equa-tions obtained by these methods.

The method of solution of the unknown tempera ture field def ined by equations (C23)to (C31) is essentially that used in reference 12 . This method is classed as analternating-direction implicit method which has the advantages of being implicit , stable,and amenable to rapid solution. With this method, the second deriva tive a2T/ay2 isreplaced by a second difference evaluated in termsf unknown temperatures at t imeT + AT, nd the other deriva tive a2T/ax2 is replace d by a second difference evaluatedin t er ms of known tem perat ure s at t ime 7. This formulation is implicit in they-direction. The procedure is then repeated for a seco nd tim e step of equal size, withthe formulation implicit in the x-direction. The alternation in solution, that is , column-row, is continued over the specified time period. It should be noted that to maintain astable numerical solution, a pair of successive row-column solutions is required. Suc-ces sive pair s of solutions, however, may have different time steps i f desired.

Equations (C23) to (C31) take the form, for either a row o r a column solution, of

where j = 1, 2 , 3 , . . .,S for he column solution and j = 1, 2 , 3, . . ., L for herowsolutions.Equation (28) rep res en ts L o r S equations and L or S unknown tem -pera tures. Since equation (28) resul ts in a trid iagonal matr ix of unknown tempera tures ,this set of equations can be quickly solved simultaneously by using a procedure based onthe Gauss elimination method. This procedure is discussed in reference 12 .

RESULTS AND DISCUSSIONIn the following section, the accuracy of the solution for the unknown tem pera ture

field defined by equations (C23) to (C31) is evaluated by comparing the results of thenumerical solution with two exact solutions. Also, the results obtained by application ofthe present analysisvia the associated computer program to several special wo-dimensional ablation cases are presen ted. Computer-drawn curves showing the com-puted results f or some of these cases illustra te t he plotting feature f the program.

Comparison With Exact SolutionsThe exact solutions to two heat-tra nsfer problem s are used t o eval uate th e acc u-

racy of the numerical results. The exact solution of an orthotropic ablating body withtemperature-dependent properties is not available. Therefore, comparisons are madewith results from exact solutions for homogeneous, nonablating bodies with constantproperties.16


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Insulated thick-walled hemisphere.- The exact steady-state temperature distribu-tion of the insulated thick-walled hemisphere shown schematically in figure4 is derivedin appendix D in spherical coordinatesand is

T(r,O) = 4 T0(3 cos28 - 1)T, +

TABLE 1.- TEMPERATURE DISTRIBUTIONS OF THICK-WALLED HEMISPHEREFROM EXACT SOLUTION AND FROM NUMERICAL ANALYSIS

0-0.162-.488-.467-.356-.209

0.125-0.220-.507-.493-.395-.248

0.25-0.234-.473-.463-.381-.254

0.375-0.240-.407-.397-.338-.245

0.50-0.243-.3 18-.304-.265-.210

0.625-0.248-.280-. 94-. 68

1 -. 43i

I 0.75-0.257-.136-.093-.067-.058

0.875 I 1.00

One-dimensional insulated slab.- The exact transient temperature distribution inaone-dimensional insulated slab shown schematic ally in figure5(a) is given by equa-tion (All) n reference 13 as (in the notation of the present paper)

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I I 1 I.1

1 ..2 .3 .4 .5

Time, se c( a ) Time step, 0.0625see.

Figure 5.- T r a n s i e nt t e m p e r a t ur e h i s t o ri e s f o r a one-dimensional nsulated s l a t .% = 1.1349 MW/m2; k = 62.35 W/m-%; p = 160.1 kg/d; cp = 0.4187 /g-%.

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The agreement between numerical and exact results for the larger time steps s notas good as that obtained with the smaller time step. T he maxim um error s for the firstthree t ime stepsof 0.0625 second were, respectively, 0.70, 0.36, and 0.25 percent, whereasthe maximum er rors f or th e larg er tim e step were -1.22, 0.56, and 0.26 percen t. In bothcases, the solutions were stable and converged rapidly.

ApplicationThe results obtained from application of the present analysis to ablating bodiesare

presented in this section. Two of the cases discu ssed, an ablat ing hemisphere-con e bodyand an ablating hemisphere, a r e of gener al interest. Two additional cases provide a com-parison, on a qualitative and quantitative basis, between test and analytical results . Allthe c ases illust rate situat ions that are ot amenable to a one-dimensional analysis.

Hemisphere-cone.~ ""ody.- A graphite hemisphere-conebody is exposed to stagnationconvective and radiant heating rates of 34 and 11 MW/m2, respectively, and a streamenthalpy of 93 MJ/kg. The se ene rgy levels are typ ica l of ear th entr y at hyperbolic veloc-iti es. The body is assumed to be subjected to this environment for 60 seconds. Fig-ure 6(a) shows computed geometry as a function of time during the 60-second exposure.

25

15

10

5

0 5 10 15 20 25 35 40= >

( a ) Shape hange.Figure 6.- Graphitehemisphere-cone body. I n i t i a ls t a g n a t i o nc o l d - w a l lh e a t i n g r a t es

of qc = 34 MW/m2 and qr = 1l MW/m2; H, = 93 MJ/kg; i n air.20


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Note the nonuniform surfac e r ece ssio n o ver thebody which require s continued adjustm entin pressure and he ating-ra te distributio ns over the body. Adjustments to the stagnationheating rates are required to account for increased nose bluntness. These latter adjust-ments, as computed with the pre sen t ana ly sis , are hown in figure 6(b).

1.4,

'80I I

20 40I I 60Time, se c

(b) Var ia t ion of s t agn a t ion hea t in g ra tes (due t o changingnose bluntness) wi th time.Figure 6.- Concluded.

A 5- by 10-node network, that is, S = 5 and L = 10, w a s used for this example.A finer network would produce sm oother profiles of the hemisphere-con e body than ar eshown in figu re 6(a). Thi s is particul arly tru e in th e hemis pherica l portio n of the body.The tot al number of stations that should b e usedwill depend on the physical size of thebody considered and the gradients of t he surface inputs, that is, heating rates and pres-su re on the surface.

Hemisphere.- A low-density phenolic-nylon hemisphere, approximated with a 10- by10-node network, is assumed to be exposed to a convective heating ra te of 3.4 MW/m2 inair for 110 seconds. The method presented in reference 7 was used to approximate thethermophysical and thermochemical properties of a charring ablator in a single-layermaterial.

Figure "(a) shows the outer surface location as a function of t ime . For the se ca l-culations heheating-ratedistributionwasbasedon Newtonian pressur edistribution,andtherefo re, fig ure 7(a) shows qualitative rather than quantitative results for an ablatinghemisphere. Note that for about the first 70 seconds, nonuniform ablation occurs acrossthe surface; however, after 70 seconds, the recession is uniform. This behavior, althoughexpected, would not be revealed ina one-dimensional analysis o r a multidimensional anal-ysis t hat do es not consider shape change. Similarly, the variation in the stagnationheating rate with nose bluntness would not be reveale d ina one-dimensional analysis or a

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2 .0

1 . 5

1.0

.5

0 - 5 1 .0 1 . 5 2 . 0 2.5 3.0 3.5z>

(a) Shape change.Figure 7.- emispherical nose capf low-density phenolic-nylon. Initial stagna-tion cold-wall convective heating ratef 3 . 4 MW/~I?;H, = 7N/kg; in air.

multidimensional analysis that does not consider shape change. The variation in thestagnation heating rate with time resulting from no se bluntness, as computed by thepresent analysis, is shown in figure 7(b).

Teflonsphere.-Theresults of a series of te st s on eflonspheres at variousheatingconditions are given in refere nce 14. One of the spheres, model E, was tested at an ini-tial stagnation cold-wall convective heating rate of 6.41 MW/m2, a stream enthalpy of4.88 MJ/kg, and a Mach number of 2.6. The experimental profile history for model E isshown in figure 8(a).

The analytical profile history for modelE, obtained by using the present analysiswith a 10- by 10-node network, is shown in figure 8(b). The agreement between the

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1.0

-9 -

(qc/qc,o)stag .a -

.7 -

.6 - - I0 I I I I I20 40 60 80 100 720

(b ) Var ia t ion o f s tagn at ion hea t ing r a te (due t o ch an g in gnosebluntness)with ime.Figure 7.- Concluded.

Approximate locationT = 0.6 sec of cen t e r i n e o f7 i n i t i a l sp here

Model E( a )x p e r imen ta lr o f i l ei s t o r y ( re f . 1 4 ) . ( b )n a l y t i c a lr o f i l ei s t o r y .Figure 8.- E x p er imen ta l an d an a ly t i ca l p r o f i l e h i s t o r i e s f o r an ab la t ing te f lon sphere .

I n i t i a lcond i t ions : qC,s tag = 6.41 Mw/m2; He = 4.88 MJ/kg;Mach number, 2.6.

2 3


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experimental and analytical profile histories is good, qualitatively , up to about 8 seconds.The analytical calculations were terminatedat 8 seconds because the present analysiscannot consider the entire sphere, but only the forward hemisphere. Therefore, in theanalysis the stagnation point cannot recede more than a length equal to one body radius.Also, the heating-rate distribution used is not accu rat e for highly blunted bodies.

A comparison between the experimental and analytical stagnation-point recessionis shown in figure 9. The analytical and experimental recessions are in good agreement.This agreement is better quantitatively than that shown in figure 8 because the inaccurateheating-rate distribution predicted with the present method for highly blunted bodies didnot affect stagnation-point recession as directly as it affected the overall recession.

1 - &stag2 *stag,o

0 Tes t data r e f . 1 4Present analysis

0

I0 10 120

Time, secFigure 9.- Comparison between the experimental and analyticalstagnation-point recession or an ablating teflon sphere.Initial conditions:

He = 4.88 W/kg; Mach number, .6.c, stag= 6.41 Mw/m2;

Laser test.- The results of tests of a low-density phenolic-nylon ablation ma te ria lat high radiant heating rates produced by a laser beam are given in reference 15. Thespecimen was exposed to a laser beam, with the energy distribution shown in figure lO(a),for 2 seconds. The maximum radiant heating rate at the cente r was about 40 MlV/m2.The final geometry of the ablated specimen was characterized by a very thin shell ofcharred m aterial, figure 10(b). The present analysis was used to determine whether24


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sufficient radiation from the sides could limit recession along the sides and thus allowthin shell of cha r to remain.

The profiles computed first by assuming no radiation from the spe cimen sides a ndthen by assuming that th e sides radiate a r e shown in figu re lO(c). Note that without radi-ation, the cusp profile is similar to that obtained during the test except that the sidesrenot as high. When the side s are allowed to radiate, the center recedes about the same

q,(laser beam)

edge e edge( a ) Energydi s t r ibu t ion .

4 b . 6 4 cm( b ) Estimatedspecimenshape.

S i d e r a d i a t i m" No s i de r a d i a t i on(c )Ca lcu la tedrecess ion .

I n i t i a l ou t e r surf ace

Figure 10.- L a se r t e s t r e su l t s f o r l a w - de ns i t y phe no l i c - ny l onspecimen ref. 15).25


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amount as with no radiation; however, tall sides, more characteristic of the experimentalresults, are obtained. This substantiates the conclusion drawn in reference 15 that thesides of the model were cooled sufficiently by radiation to permit a considerable height ofchar to remain at the edges.

The pro fil es shown in figure 1O(c) were computed with a 10-by 5-node network(i.e., S = 10 and L = 5). The computation was repeated with a 10- by 10-node networkwith essentially the same results.

CONCLUDING REMARKSThe differential equations governing the transient response of an ablating axisym-

metr ic, ortho tropi c body have been derived for fixed points.in a moving coordinate systemThese equations have been expanded into finite-difference form and have been programedfor numerical solution with an implicit technique on a digital computer. Numericalresults compared favorably with the exact solutions for simplified conduction problems.

The determina tion of the changes in the body geome try as ablation occurs, and theeffec t of these changes on the surface energy inputs, is a salient featu re of this analysis.This feature was used to obtain satisfactory agreement between numerical and experi-men tal result s of an ablating teflon sphere and small test specimen exposed to high-intensity laser beam.Langley Research Center,

National Aeronautics and Space Administration,Hampton, Va. , April 2 , 1971.

26


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APPENDIX A

SHAPE CHANGE DUE TO MASS LOSSMass transfer at the surface due to ablation causes a change in material thickness.

The change in ma terial thickn ess at any point n on the sur fac e is, from equation (18),

When there are va riations in he ating and pressure ov er the ody surfa ce, the mas s tra ns-fer also varies over the surface. This variation in mass transfer cause s a nonuniformchange in thickness and, hence, a change in shape which consequently alters the heati ng-rate and pressure distributions over the surface.

The heating-rate and pressure distributions are recalculated, with the anal ysis ofreference 11, as the shape changes. The m ethods used to evaluate the shape parametersWn, qn , and Rstag requir ed to implement the analysis of reference 11 a r e given in thisappendix.

The w,z coordinate system shown in figure 2 is used to define the surface geometryat any time. The surface coordinates at station n as a function of time are

andZ ( T ) = z ( T ) ~ 6 ( ~ ) ~6 ( ~ ) in 8 (A3

The instantaneous value of the angle Qn between the free-stream velocity vectorand the local normal to the surface s requ ired in defining a new pressure distribution.This angle is determined as follows:For n = 1,

for 2 S n < L,


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APPENDIX A - Concludedand for n = L,

Both the stagnation convective and radiative heating rates are functions of theinstantaneous radius of cu rvature. . The radius of c urv atur e in the stagn ation re gionsobtained by f inding the radius of a circle passing through points (S,1) and (S,2) (seefig. 11) in w,z coordinates.

Figure 11.- Location o f f i n i t e - d i f f e renc e s t a t i o ns .The equation of a circle in the w,z coordinate system with its cent

is of the form

Evaluating equation (A7) at the point (S,2) and solv ing for RStag givescurvature in the stagnation regionas

28

!r on the z-axis

(A71he radius of


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APPENDM BDERIVATION OF THE DIFFERENTIAL EQUATIONS AT

THE COORDINATE SINGULARITYThe governing differential equation, equation (3), has a line of singularity at x = 0.

This singularity can be eliminated by examining the behavior of the scale factor h3 asx - 0.

AS x - 0, Rcyl - x and cos 8 =-cyl -- Therefore,R R'h3 = Rcyl + y cos 8 - x 1 +x =( R) Xhl(3x-oh l

and ($) Xx-0

Now consider equation (3) in the expanded form

As x - 0, equations (Bl), (B2), and (B3) reduce equation (B4) oAt x = 0, the axisymmetric-body assumption requires that - 0 and -

As a result , the second termon the left side of equation (B5) becomes indetermina te.Applying L'Hosp ital's rule to this term gives

aT a h l 0.ax ax

29


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APPENDIX B - ConcludedEquation (B5)may now be written as

which is the governing differential equation along the line x = 0 and is solved with theboundary conditions of equations (19) nd (22).

30


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APPENDIX CFINITE-DIFFERENCE EQUATIONS

The differential equations are put in finite-difference form through the useofTaylo r's series expans ions. Forwa rd, central , and backward differences are used. Themethods used to obtain these differences are from reference 8. These methods, in gen-eral, useTaylor's series expansions at points *At o r a~oevaluatefirst-orderderivativesandTaylor's series expansions at points *A5/2 or *A77/2 to. evalua tesecond-order derivatives. Typical finite-difference expressions used are summarizedin this appendix. The ske tches in figure 12 illustrat e the spatial relatio nship betweenpoints used in the Taylor's series expansions.

( m , x )

(a)For standarderivatives.b) For cross derivatives.Figure 12.- Spatial relation between pointssed in Taylor's series expansions.

First-Order DerivativesThe firs t-or der der ivat ives are given by the following equations which a re co rre ct

to orde r of Aq2:Forward difference

Central difference


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APPENDIX C - ContinuedBackward difference

Second-Order DerivativesThe second-order derivatives are given by the following equations:

Forward difference

whereB = hlhgk.Central difference

Backward difference

term

Term

Cross DerivativesThe methods used to evaluate cross derivatives can behown by considering the

(C7) is first expanded in the q-direc tion. T hus,

32


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APPENDIX C - ContinuedThe first derivatives are then expanded in the x-direction to give

Equation (C9) is correct to A$, Ax2. Equations (C8) and (C9) are cen tra l-d iff ere nceexpansions.

The forward expansion of te rm (C7) is obtained by evalua ting at Aq and 2 A q ,which gives

The first derivatives are thenexpanded to give

The backward expansion of te rm (C 7) is obtained by evaluat ing at -Aq and -2 Aq.The resulting expression is

A Special CaseTheboundariesalongx = 0 andx = L require special attention. At the se two

locations the boundary conditions are, respectively,

and

3 3


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APPENDIX C - ContinuedBecause of symmetry at x = 0,

Therefore, at x = 0, the boundary condition becomes

A forward-difference expansion different from equation (C8) can now be obtainedfor the first te rm on the left side of equation (25). This diff erence exp ress ion of a2T/ax2is obtained as follows:A Taylor's series expansio n of Tm ,2 in ter ms of T m, l

Similarly, heexpansion of Tm,3 n erm s of T m , l is

Eliminating the third deriva tive between equations (C 17) and (C 18) give s

After the boundary condition of equation (C16) is utilized, equation (C19) becomes

Along x = L, it is advantageous to write the first te rm in equation (3) in a semi-expanded for m, which is , in the 6 ,q coordinates,

34


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APPENDIX C - Continuedwhen a half-sta tion backward-di fference expression (eq. (C10)) is used, the first term onthe r ight side of equation (C21) becomes

The last te rm is evaluated from the boundary condition at x = L (eq. (C14)). Theother terms are expressed in forward-, backward-, or central-difference form asrequired.

Finite-Difference Approximations of the F ield EquationsThe locations of the finite-difference stations (m,n) are shown in figure 11. The m

and n subscripts correspond to the r-coordinates and F;-coordinates, respectively.The methods used to change the differential equations to finite-difference form a r e

given in the preceding sections of this appendix. A summary of the finite-difference equa-tions obtained by these methods is presented here.

Fo r any point (m,n), where 1 < m < S, 1 < n < L, the govern ing equation (eq. (9)) isp m y n + l - Tm,n> - (T)3k5 p m , n -

m p + - 12 m,n- -2+ &Flh3$)nAV m+--,n Tm+l, n - Tm,n) tlh 3$) (Tm,n - Tm-l ,n2 m- 3;'" 3

(Equation continued on next page)35


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APPENDIX C - Continued

At station (m = 1, n = l ) , he boundary condition (eq. (26)) is combined with equa-tion (25) to give

For the curvilinear coordinate system, the a t e rm is set equal to 2. It should be notedthat for Cartesian coordinates,a slight modification must be made to equations (C24),(C25), and (C26). This modif ication is required since there is no line of s ingular ity in theCartesian system as there is in the curvilinear system. The correct forms of equa-tions (C24), (C25), and (C26) for the Cartes ian system are obtained by sett ing the a termin these equations equal to 1.

For station (1 < m < S, n = l ) ,equations (24) and (25) yield



39/46


For station (m = S, n = l), quations (25) and (27) result in

For station (m = 1, 1 < n < L), equations (9) and (26) combine to give

For station (m = S, 1 < n < L), combining equations (9) and (20) yields :

+



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S-2 ,n +l

(C28)For station (m = 1, n = L), combining equations (9), (22), and (23):gives



41/46


at1,1= (hlh3Pcp) (C29)1,1

For stations (1 < m < S, n = L), equations (9) and (23) yield

m+l,L - Tm-1,L

+ - 1 2) Plh3%) Tm+l,L - Tm,L) - (hlh3%) (Tm,L - Tm-l,L)2 1A7 m,L m+-,L m- -,L2

= (hlh3Pcp) m,L ["L A r + (%) (Tm+l,L'P m,L 2 Av

39


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APPENDIX C - ConcludedFor station (m = S, n = L), equations (9), (20), and (23) are combined to yield

3 T ~ - 2 , ~4 T ~ - 2 , ~ - 14xb

- 4 T ~ - 1 , ~ - 1T ~ - 1 , ~ - 24Xb A t )-(%)- -,L (T"L -2

r 77

" 3 A:262) 1 3%) s-z,L(TS-1,L - TS-2,L) - 9(hlh3%) 1 (TS,L - TS-1,L)S,L s- ,L2

40


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APPENDIX DDERIVATION OF EXACT SOLUTION

The governing differential equation fo r the steady-state temperature distribution inan internally insulated, thick-walled, hollow hemisphere shown schematically in figure 4is

The boundary conditions are

and

The general solution to equation (Dl) y the method of s epar ation of var iab les is

where Ao, Bo, Co,Do,An,Cn, nd Dn areco ns ta nt s of integration nd Pn(COS e)is the Legendre polynomial.

When the boundary cond itions, equations (D2) and (D3), are used to evaluate thecoefficients in equation (D4), the solution to equation (Dl) is

4T(r,O) =3To +

41


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REFERENCES1. Peterson, W. D.; and Spanier, J.: HOT-2: A Two-Dimensional Transient Hea t Con-

ducti on Program for the CDC-6600. WAPD-TM-669, U.S. At. Energy Comm.,June 1967.2. McClure , John A,: TOODEE - A Two-Dimensional, Time-Dependent Heat ConductionPr og ra m. AEC Res. and Develop. Rep. IDO-17227, U.S. At. Energy Comm., April1967.

3. Fri edman , H. A.; and McFarland, B.L.: Two-Dimensional Transient Ablation andHeat Conduction Analysis for Multimaterial Thrust Chamber Walls. J. SpacecraftRockets, vol. 5, no. 7, July 1968, pp. 753-761.

4. McCuen, Peter A.; Schaefe r, John W.; Lundberg, Raymond E.; and Kendall, Rober t M.:A Study of Solid-Propellant Rocket Motor Exposed Materials Behavior. AFRPL-TR-65-33, U.S. Air Fo rce, Feb. 26, 1965.

5. Popper , L. A.; Toong, T. Y.; and Sutton, G. W.: Three-Dimensional Ablation Con-side ring Shape Changes and Internal Heat Condition. AIAA Paper No. 70-199,Jan. 1970.

6. Moyer, Car l B.; Anderso n, Lar ry W.; and Dahm, Thomas J.: A Coupled ComputerCode for the Transien t Therm al Res ponse andblation of Non-Charring HeatShields and Nose Tips. NASA CR-1630, 1970.

7. Thornton, William A.; and Schmit, Lucien A., Jr.: The Structural Synthesis of anAblating Therm ostructura l Pan el . NASA CR-1215, 1968.

8. Swann, Robert T.; Pittman, Claud M.; and Smith, James C.: One-DimensionalNumerical Analysis of the Transie nt Respon seof Thermal Protection Systems.NASA TN D-2976, 1965.

9. Scala , Sinclaire M.; and Gilbert , Leon M.: Sublimation of Graphite at HypersonicSpeeds. AIAA J., vol. 3, no. 9, Sept. 1965, pp.1635-1644.

10. McCarthy, J. F., Jr.; and Hanley, G.M.: Ea rth En try at Hyperbolic Velocities.AIAA Pa pe r No. 68-153, Jan. 1968.

11. Lees, Lester: Laminar Heat Transfe r Over Blunt-Nosed Bodies at Hypersonic FlightSpeeds. Jet Propulsion, vol. 26, no. 4, Apr. 1956, pp. 259-269, 274.12. Gavril, Bruce D.; and Lane, Frank: Finite Difference Equations and Their Solution

for the Transient Temperature Distribution in Composite, Anisotropic, GeneralizedBodies of Revolution. Tech. Rep. No. 230 (Contract No. NOrd 18053), Gen. Appl.Sci. Lab., Inc., May 26, 1961.

42


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13. Brooks, William A., Jr.: Temperature and Thermal-Stress Distributions in SomeStructural Elements Heated at a Constant Rate. NACA TN 4306, 1958.

14. Howell, Robert R.: An Experimental Study of the Behavior of Spheres Ablating UnderConstant Aerodynamic Conditions. NASA TN D-1635, 1963.

15. Brewer, Will iam D.: Ablative Material Perf orma nce in High Radiative Heat FluxEnvironment Produced by a Carbon Dioxide Laser . AIM Paper No. 70-864,June-July 1970.

NASA-Langley, 1971- 3 L-7474 43


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