APRIL 1989 TECHNICAL NOTES 225

Table 1 Comparison of results (Mach 20, 20 km)Normalized distance from wall rn&

0.0 0.50 1.0Present Ref. 18 Present Ref. 18 Present Ref. 18

533.4 536.9 527.4 532.3 506.6 511.7

11.91 11.85 11.86 11.76 11.40 11.37T(K)

Molefraction

002NN2NOe~

7283

0.29511.089-30.19670.48512.133-23.927-4

7395

0.29339.265-40.20030.47762.070-23.945-4

7252

0.29571.123-30.19150.48942.143-23.821-4

7388

0.29289.232-40.19960.47802.065-23.921-4

7243

0.29581.091-30.19310.48812.106-23.828-4

7357

0.29339.232-40.19700.48012.057-23.840-4

*rn' - 0 at the wall; rn = 1 at the shock.

14

12

10

8

V 6

4

2

-2a) Normalized density

141

12

10

8

V 6

4

2

0

-2 b)8 10 12 14- 2 0 2 4 (

x

b) Normalized temperatureFig. 4 Hypersonic inlet, Mach 20.

The temperature along much of the forebody surface is inexcess of 7000 K. Figure 4 shows equilibrium flow through ahypersonic inlet at the same conditions. This problem waschosen to demonstrate the code's ability to compute internalas well as external flows. Only the upper half of theaxisymmetric inlet is shown. Some interesting flow phe-

nomenon is apparent, including a Mach stem and a reflectedshock. No significant problems were encountered when thecode was applied to these more complex configurations.

ConclusionsThe objective of this work was to develop an algorithm that

could accurately predict inviscid, equilibrium, high Machnumber flows. That goal has been achieved. The method hasproved its ability to capture strong shocks. Solutions for ageneric axisymmetric hemisphere at Mach 20 compare wellwith previously published results. The code has shown itsability to be applied to both internal and external hypersonicflows over a variety on configurations.

References'Gnoffo, P. A., "Application of Program LAURA to Three

Dimensional AOTV Flowfields," AIAA Paper 86-0565, Jan. 1986.2Balakrishnan, A., Davy, W. C., and Lombard, C. K., "Real-Gas

Flowfields about Three Dimensional Configurations," Journal ofSpacecraft and Rockets, Vol. 22, Jan.-Feb. 1985, pp. 46-53.

3Eberhardt, S. and Palmer, G. E., '"A Two Dimensional TVDScheme for Inviscid, High Mach Number Flows in ChemicalEquilibrium," AIAA Paper 86-1284, 1986.

4Steger, J. L. and Warming, R. F., "Flux Vector Splitting of theInviscid Gasdynamic Equations with Application to Finite DifferenceMethods," NASA TM-78605, 1979.

5Gordon, S. and McBride, B. J., "Computer Program for Calcula-tion of Complex Chemical Equilibrium Compositions, Rocket Perfor-mance, Incident and Reflected Shocks, and Chapman-JouquetDetonations," NASA SP-273, 1976.

6Steger, J. L., "Implicit Finite-Difference Simulation of Flowabout Arbitrary Two Dimensional Geometries," AIAA Journal,Vol. 16, July 1978, pp. 679-686.

7Beam, R. M. and Warming, R. F., "An Implicit Factored Schemefor the Compressible Navier-Stokes Equations," AIAA Journal, Vol.16, April 1978, pp. 393-402.

8Stull, D. R. and Prophet, H., JANAF Thermochemical Tables,2nd ed., National Standard Reference Data System NBS37, June1971.

9Nakahashi, K. and Deiwert, G. S., "A Self-Adaptive-Grid Methodwith Application to Airfoil Flow," AIAA Paper 85-1525, July 1985.

10Lyubimov, A. N. and Rusanov, V. V., "Gas Flows Past BluntBodies, Part II: Tables of the Gas-Dynamic Functions," NASATT-F715, 1973.

chkL

Deforming Grid Method Appliedto the Inverse Problem

of Heat Conduction

R. C. Mehta* and T. JayachandrantVikram Sarabhai Space Centre,

Trivandrum, India

Nomenclature= heat capacity of the material= heat-transfer coefficient= thermal conductivity= thickness of material

Received Dec. 24, 1986; revision received Sept. 30, 1987. Copyright American Institute of Aeronautics and Astronautics, Inc., 1987. Allrights reserved.

* Engineer, Aerodynamics Division.fEngineer, Solid Motor Project.

226 J. THERMOPHYSICS VOL. 3, NO. 2

qw = surface heat fluxT = nodal temperatureTaw = adiabatic wall temperatureT0 = initial condition for temperaturet = time variableX = Cartesian coordinate with respect to fixed referencea = thermal diffusivityAr = computing time intervald = heat penetration depthp = material density(j)j = finite-element shape functions

Introduction

IN the inverse heat-conduction problem,1 the prediction oftransient wall heat flux is accomplished by inverting thetemperature history as measured by a thermocouple located atthe inner or outer surface of the solid material. Thefinite-difference and finite-element methods2'3 have been thepredominant numerical techniques for the solution of theinverse heat-conduction problem. A potential difficulty withnumerical techniques in the solution of the inverseheat-conduction problem is the lack of a priori informationconcerning the gradient in the dependent variable to beestimated. Therefore, a predetermined distribution of nodalpoints is either inadequate or computationally uneconomical.What is needed is a method wherein the distribution of nodalpoints is to be adjusted in response to the nature of thecomputed solution.

Lynch and O'Neill4 have used continuously deforming finiteelements in space and a finite-difference scheme for theintegration in time to solve the Stefan problem. This methodhas great intuitive appeal, provides a rationale and a mechan-ism for generating difference equations on a nonuniform grid,and also allows the use of higher elements in regions where theyare suited to the physics of the problem. In the absence of griddeformation, the method reduces exactly to the conventionalGalerkin finite-element procedure.5 The particular method ofsolution in time may still be specified by the user to suit thedetail of the specific problem; it is not dictated by the movingmesh feature. *

The purpose of the present Note is to analyze an inverseheat-conduction problem using a finite-element and deforminggrid technique in conjunction with measured thermocoupletemperature at the outer surface of the finite slab. An explicitdeformation pattern via a heat penetration depth concept6 isemployed for adequate modeling of the transient steep gradientand the time delay in temperature response resulting from thesurface and thermocouple location. The proposed scheme isshown to be accurate, efficient, and convenient.

AnalysisConsider the one-dimensional heat-conduction equation

with constant thermophysical properties of material. Thephysical problem is that of a slab of finite thickness exposed toa transient heat input at one surface and perfect insulation atthe other. For the system under consideration, the governingheat-conduction equation is

s\ ^ T p 'T'

with the boundary conditions

, dT(09t)dX

8T(L9t)= qw> t>v

= 0, t > 0dX

and the initial condition

T(X90) = r0 for all X

(1)

(2)

(3)

(4)

Deforming Finite-Element FormulationIt is appropriate to discretize the diffusion equation in terms

of deforming finite elements in order to treat the transient steepgradient of wall heat flux near the boundary of the heatedsurface. The spatial shape functions j in the Galerkin weakform of the equations will be time-dependent at every fixedlocation X, and can be written as

T(X,t) = j = 1,2,3, . . . ,7 (5)where the 7} are the N nodal values of the temperature actingas generalized coordinates of the problem. The time derivativeof the temperature is accordingly discretized as

dT dj~

(6)

and implies, contrary to conventional fixed-grid formulation,the time rate of change of the shape function. Using theisoparametric element and the Langrangian nature of thefinite-element approximation, the term dfyldt is evaluated as7

(7)

The term dX/dt is the mesh deformation velocity and iscontinuous throughout the spatial domain. Assembling Eqs.(6) and (7) yields

dTdt ''

dX (8)

The last term in the Eq. (8) can be viewed as a transport termaccordingly for the convective effect of mesh motion.

Following the usual Galerkin procedure, one obtains theheat-conduction equation in the form of a system ofsemidiscretized differential equations in _terms of nodaltemperature, collected in the global vector T as

(9a)where a_superscript dot denotes time partial differentiation,and K, Kd, C, and g are the conventional thermal conductivitymatrix, the convective matrix accounting for meshdeformation, the heat capacitance matrix, and the thermal loadvector, respectively, with typical components of the form

^r,Jo

j = 2,3,. . . ,7V (9b)All of these characteristic matrices depend on time and the

unknown surface heat flux qw. Conventional heat-conductionfixed-grid formulations are simply obtained by omitting theconvective matrix Kd in Eq. (9). Equation (9) is now integratedin time, using a modified Galerkin scheme4 at time t* as

/* = *+** (10)

(H)where the subscripts denote the corresponding time station.The value of e is taken as 2/3. This algorithm possesses

APRIL 1989 TECHNICAL NOTES 227

Table 1 Wall heat flux at various grid arrangements

M123456789

10111213141516

r dtA/ atouter

surface, K

300300300300300326342356380402425440460479507528

Uniform grid

T;,K658.3763.4883.7971.5

1053.91054.61028.91067.51097.41132.51165.91198.91231.21262.91238.51252.9

qw x 106,W/m2 '

3.71553.71553.71553.71553.71553.71552.70082.69822.70452.70472.70382.69082.68292.67282.09412.0856

h, W/m2K

1624.11702.31801.61881.71963.91964.51408.81436.91463.01491.41518.91539.71564.61588.11226.41231.8

Nonuniform grid

T;,K671.0765.2892.9978.6

1063.71067.91046.41091.81124.41163.31178.71231.51263.41293.61253.81264.6

qw x io6,W/m2

3.84623.84623.84623.84623.84623.84622.84862.83992.85962.85832.85262.80542.77582.73832.01471.9815

h, W/m2K

1690.61763.41873.41954.52043.42044.91499.61531.61569.81603.31632.61636.21649.71657.11190.61178.5

T;,K745.3877.2

1051.11111.61072.91074.01049.51081.91115.41152.21188.11221.81255.11286.91253.01266.7

Deforming grid

qw x IO6,W/m2

4.51734.51734.51734.51734.51734.51732.81802.82012.84252.84632.84512.81172.79102.76402.09152.0675

/jfl = heat-transfer coefficient (Bartz).11

h, W/m2K

2052.72183.52339.52462.52411.62413.11485.91512.81552.81586.71618.51630.81650.61665.91235.41231.6

., 10 TU^r. \o.nAc

/^,W/m2Ka

2254.22254.22254.22254.22254.22254.22254.22254.22254.22254.22254.22254.22254.22254.22254.22254.2

NPD

MOVING GRID,t-U

NONUNIFORM GRID

.002 -004 .006 O08 .Ofl -OI3UNIFORM GRID

Fig. 1 Finite slab discretization.

unconditional stability and accuracy and is also free fromoscillatory behavior.8 But in Eq. (11), the surface heat flux qwis an unknown parameter; thus, the solution of the completeproblem from X = 0 to XM cannot be readily obtained becausethe boundary condition is not known but, rather, thetemperature history is known at an interior point XM. Inestimating, one minimizes

(12)

where Tc and TM are, respectively, the calculated and measuredthermocouple temperature at (XM,t). The above-mentionedminimization scheme is found suitable when a singlethermocouple is used to measure the temperature history andalso a larger time step in the computation is employed in orderto satisfy the larger time interval of measurement. Thecalculated temperature is, in general, a nonlinear function ofqw. Hence, to predict the unknown surface condition, aniteration is required in this method, which is a disadvantage. Asimple procedure approximates at each iteration the calculatedtemperature by a Newton-Raphson iteration procedure9 (withquadratic convergence). This iteration scheme begins with aninitial value of qw and continues until, say, |F| < 10~4.

Deforming Grid StrategyIt is observed that the temperature response at an interior

location or outer surface is delayed and damped with respect to

difficulties in determining the unknown wall heat flux. It stemsfrom boundary condition (2) that the correct estimation of theqw will depend heavily on the ability of the deforming gridtechnique to represent accurately the interior gradient near thesurface. In many aerospace applications, such as re-entry androcket nozzle heat-transfer problems, the sharp fronts in thetemperature distribution are found near the heated surface;therefore, such a steep thermal gradient must be dealt withusing the deforming grid technique. A fine mesh in the vicinityof the boundary within the zone of a heat penetration depth6allows for a suitable and accurate solution to this problem.This analysis included NPD elements of^he initial (uniform ornonuniform) grid in a zone 6(f) = (at)* corresponding to theinstantaneous heat penetration depth. The NPD elements aredilated, and the remaining ones are contracted using the fol-lowing instantaneous coordinates locations:6

(13)

where m = Npr>, Xm(i) = 6(i) and X? are the initial node posi-tions with respect to fixed origin. The deforming grid strategyis carried out in the preceding formulation as long as no tem-perature rise from the initial temperature is observed by thethermocouple. Once this delay in the measurement of tempera-ture history is over, the deforming grid strategy is stopped andthe solution continues with that grid. The heat penetrationdepth is an explicit function of time.

ExampleMoving-grid finite-element formulation, discussed in the

previous sections, has been utilized for estimation of the wallheat flux and the surface temperature at the inner surface for atypical divergent rocket nozzle made of mild steel in conjunc-tion with the experimentally measured outer surface tempera-ture data in a static test. The nozzle conditions and materialproperties taken are: L = 0.0211 m, TQ = 300 K, Taw = 2946 K,thermal conductivity (average) = 35 W/m2K, p = 7900 kg/m3,c = 545 Ws/kg K, burning time = 16 s. The heat input con-sists of a convective heat flux at the inner surface (X 0). TheX = L is insulated. An insulated chromel/alumel thermocoupleof 30 gage was used for measuring the temperature history atthe outer surface. A time delay of 6 s was noticed in the mea-surement of temperature history.

228 J. THERMOPHYSICS VOL. 3, NO. 2

In the determination of wall heat flux, three different types ofgrid arrangements (namely, uniform, nonuniform, and de-forming grids) were used in conjunction with the above-men-tioned Newton-Raphson iteration procedure. An iterativeindex is included in the computer program to count the numberof iterations necessary to achieve desired accuracy in the mini-mization of the function \F\. In each case, 11 nodes and a timeinterval of 1 s are taken for computational purposes. In thecase of nonuniform grid, 5 meshes are stretched near the con-vectively heated surface of the slab while the remaining meshcovers the rest of the region. Figure 1 shows the mesh move-ment at each time interval using the heat penetration depthconcept. The numerical computations have been performed ona CDC 170/730 digital computer. The convective heat-transfercoefficient can be calculated as

h = qw/(Taw Ts) (14)where Ts is the surface temperature. Table 1 depicts predictedvalues of wall heat flux, surface temperature, and convectiveheat-transfer coefficient using the three different kinds of gridarrangements. It can be seen from the table that for / < 6 s, theheat-transfer coefficient obtained using the deforming gridstrategy is in good agreement with the calculated results ofBartz.11 Thus, the current numerical analysis shows that themoving-grid finite-element formulation gives an adequatemodeling of the transient steep gradient of wall heat flux nearthe boundary of the heated surface. But for t > 6 s, fixed anddeforming grid results tend to coincide since spatial gridsbecome identical, as shown in Fig. 1. The estimated values ofheat-transfer coefficient calculated using three different type ofgrid arrangements are somewhat lower than the calculatedresults of Bartz for t > 6 s. The predicted value of the heat-transfer coefficient shows that the Bartz equation gives aconservative estimation of the heat-transfer coefficient. This isalso demonstrated in the previous analysis.12 The maximumnumber of iterations per step are 11 and 8 for the uniform andnonuniform grids, respectively, whereas it is reduced to 4 itera-tions per step in the case of deforming grid. This example thusreveals that the deforming grid is economical in terms of CPUtime compared to the other type of grid arrangements. Thisexample also demonstrates that the arbitrary mesh deforma-tion can be conveniently accommodated in the solution to theinverse problem of heat conduction. It is also importantto mention here that geometrical conservation is maintainedusing nonuniform and deforming grids in the analysis becausethe numerical computations are carried out in the physicalplane.

ConclusionsA mesh coupling thermal conductivity matrix with deform-

ing finite-element formulation with heat penetration depth con-cept is found useful to treat the initial time delay intemperature response. The deforming grid procedure hasshown itself to reduce substantially the number of iterationsper step to achieve the required tolerance in the analysis of theinverse heat-conduction problem.

References'Beck, J. V., Blackwell, B., and St. Clair, C. R. Jr., Inverse Heat

Conduction Problem, Wiley, New York, 1985.2Mehta, R. C., "An Efficient Numerical Method for Solving Inverse

Heat Conduction Problem in a Hollow Cylinder," AlAA Journal, Vol.22, June 1984, pp. 860-862.

3Bass, B. R., "Application of the Finite Element Method to theNonlinear Inverse Heat Conduction Problem Using Beck's SecondMethod," Journal of Engineering for Industry, Vol. 102, May 1980, pp.168-176.

4Lynch, D. R. and O'Neill, K. "Continuously Deforming FiniteElements for the Solution of Parabolic Problems, With and WithoutPhase Change," International Journal for Numerical Methods in Engi-neering, Vol. 17, 1981, pp. 81-96.

5Zienkiewicz, O. C., The Finite Element Method in Engineering Sci-ence, McGraw-Hill, London, 1971.

6Hogge, M. and Gerrekens, P., "Steep Gradient Modeling in Diffu-sion Problems", Numerical Method in Heat Transfer II, edited by R.W. Lewis, K. Morgan, and B. A. Schrefler, Wiley, New York, 1983, pp.73-93.

7Lynch, D. R. and Gray, W. G., "Finite Element Simulation ofFlow in Deforming Regions," Journal of Computational Phyics, Vol.36, July 1980, pp. 135-153.

8Owen, D. R. J. and Damjanic, F., "The Stability of NumericalTime Integration Techniques for Transient Thermal Problems withSpecial Reference to Integration Effects," Numerical Methods in Ther-mal Problems, Proceedings of the 2nd International Conference, editedby R. W. Lewis, K. Morgan, and B. A. Schreffer, Pineridge Press, 1981,pp. 487-505.

9Mehta, R. C., "Numerical Solution of Nonlinear Inverse HeatConduction Problem with a Radiation Boundary Condition," Interna-tional Journal for Numerical Methods in Engineering, Vol. 20, 1984, pp.1057-1066.

10Beck, J. V., "Nonlinear Estimation Applied to the Nonlinear In-verse Heat Conduction Problem," International Journal of Heat andMass Transfer, Vol. 13, April 1970, pp. 703-716.

11 Bartz, D. R., "A Simple Equation for Rapid Estimation of Rocket

Nozzle Convective Heat Transfer Coefficients," Jet Propulsion, Vol.27, Jan. 1957, pp. 49-51.

12Mehta, R. C., "Solution of the Inverse Conduction Problem,"AlAA Journal, Vol. 15, Sept. 1977, pp. 1355-1356.

Approximate Method for TransientConduction in Arbitrarily Shaped

Solids with a Volumetric Heat Source

K. Taghavi* and R. A. AltenkirchfUniversity of Kentucky, Lexington, Kentucky

Introduction

T HE lumped-heat-capacity technique has long been usedto predict transient thermal behavior of solids when theBiot number is small compared to unity.1 A small Biot numberimplies a negligible spatial temperature variation within thesolid. This method, when applicable, is simple and powerfulbecause it may be applied to any geometry.

In this work, a quasi-steady-state model is considered forsolids with internal heat generation and arbitrary Biotnumber. It is assumed that the temperature profile in the solidat any instant in time is similar to that of the steady state. Thebasis of this approach has been explained by Arpaci2 andparallels that for thermally fully developed internal flows. Afully developed regime has been identified for unsteadyconduction in solids with convective boundary conditions, butwithout heat generation.3 Here, we consider the presence of aheat source, the steady state becomes analogous to fullydeveloped internal flow, and the temporal behavior of thetemperature field is contained in the behavior of the mass-averaged (bulk) temperature. The approach will be shown tobe more accurate than the lumped-heat-capacity techniqueand is easily applied to solids with arbitrary shapes.

Received April 10, 1987; revision received Jan. 29, 1988. Copyright 1988 by K. Taghavi. Published by the American Institute ofAeronautics and Astronautics, Inc., with permission.

*Assistant Professor, Department of Mechanical Engineering.fProfessor, Department of Mechanical Engineering; currently at

Department of Mechanical and Nuclear Engineering, Mississipi StateUniversity.