adaptive computational methods for chemically … · and radiative effects. applicable to general...

Jnt. J. ElIgllg Sci. Vol. 26. No.9. pp. 959-992. 1988Printcd in Greal Britain. All rights reservcd

oo2O-1225{88 S3.00 + 0.00Copyrighl © 1988 Pcrgamon Press pic

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ADAPTIVE COMPUTATIONAL METHODS FORCHEMICALLY-REACTING RADIATIVE FLOWS

JON M. BASSi and J. TINSLEY ODEN2

1 The Computational Mechanics Company. Inc .. Austin, Texas. U.S.A.2nCOM, The University of Texas at Austin. U.S.A.

Abstract-In this paper. several new developments in chemically reactive computational fluiddynamics arc discussed which could have impact on future studies on chemically-reacting. viscous.compressible flows. These include: (I) The development of very fast adaptive schemes that arcvirtually geometry- and coordinate-independent and which efficiently cope with the difficult datamanagement problems associated with the evolution of mesh topologies and general algorithmstructure. (2) The development of fast a posteriori error estimation techniques that not only provide abasis for adaptation of the algorithm structure, mesh size. and local approximation order. bul alsogive an indication of the quality of a complex numerical calculation. (3) The development of aworking experimental code for the lwo-dimensionaltransient Navier-Stokes equations with chemistryand radiative effects. applicable to general geometries and boundary conditions. and designedspecifically for applications to chemical lasers.

1. INTRODUCTION

Modern computational tluid dynamics has had a profound impact on analysis and design ofmany engineering systems. A good computation model of flow events can allow the analyst tosimulate the response of a system to changes in many parameters without going throughexpensive laboratory tests. Moreover, a reliable computational model can also be extremelyuseful in designing experiments for real systems.

The main disadvantages in computational models are: (1) the reliability of the computedresults (can they be trusted as a basis for design considerations?), and (2) the expense inobtaining a model with sufficient sophistication to capture the details of the flow of interest.

The difficulties are particularly severe in the analysis of chemically-reacting flows. In suchsimulations, one often finds irregular flow geometries coupled with complex physics andchemistry, with the result of very large blocks of computing time being required for simplesimulations.

The present study focuses specifically on these fundamental problems. In particular, weconsider (1) tbe question of reliability, as one of determining a posteriori (after each time stepor at the conclusion of a calculation) estimates of the actual error inherent in the computation;(2) an optimum (nor near optimum) use of computational resources, achieved through the useof adaptive refinement of the computational mesh so as to reduce local errors in a rational way;and (3) the development of new algorithms for time-accurate viscous chemically-reactingcompressible flow simulations on dynamically evolving unstructured meshes.

Various approaches have been taken to analyze the complex physical phenomena occurringin chemically-reacting flows. The particular applications of interest here are primarily thoseclasses of flows encountered in the nozzle and cavity regions of chemical lasers. These includeanalytical [1-6] as well as numerical [7-19] approaches in which varying levels of complexityare considered. Models for one-, two- and quasi-three-dimensional tlow fields have beenproposed for laminar and/or turbulent mixing with optical properties ranging from a simpleinfinitesimal Fabry-Perot cavity [2] to general optics conditions [11].

The analytical studies reported in the literature have, for the most part, been one-dimensional in nature and include a number of simplifying assumptions which allow closedform solutions to be obtained. Among the first of these was the quasi-one-dimensional modelof Emanual [1], where mixing effects were ignored and transverse flow properties wereassumed uniform. A more elaborate model was developed by Hofland and Miraies [2,3] wherelaminar mixing occurred in a constant pressure environment with chemical reactions limited bya flame sheet approximation. This work was subsequently extended by Mirales et al. [4,5] to

959

960 J. M. BASS and J. T. ODEN

include turbulcnt conditions with arbitrary mixing rates and chemical coefficients. A somewhatdifferent approach was persued by Broadwell [6] which eliminated the flame sheet assumptionbut introduced a scheduled mixing model for turbulent conditions.

Due to the complexity of the reacting flow problem, analytical studies providc, in general,only qualitative information in a highly idealized situation. Quantitative results, for real lasingproblems, must therefore be obtained primarily from numerical investigations. While severalone-dimensional approximations have been considered by Turner [7], Emanuel [8], andPatterson et al. [9, 10], they are also generally inappropriate for describing the inherenttwo-dimensional mixing that occurs in the laser cavity region.

A number of two- and quasi-three-dimensional reacting flow models have becn presented inthe literature. Thcse models may generally be classified into two groups. The first group ofmodels, which also appeared first chronologically, employs the boundary-layer equations in oneform or another (see, for example, the work of King and Mirales [12], Quan [11], Zelany [13],and Tripodi [14]). These models are restricted by the usual limitations inherent in theorder-of-magnitude, boundary-layer assumptions and are not completely satisfactory forhandling certain nozzle configurations and the pressure mismatches occurring at the nozzle exitplane. The other group of models incorporates either the shear-layer equations or theNavier-Stokes equations as seen in the work of Thoenes and Hendricks et al. /15, 16],Ramshow and Dukowicz [17], Rapagnani and Lankford [18], and Baev [19]. These models aresomewhat morc complex than the boundary-layer exprcssions and incorporatc the importanttransverse prcssurc gradient effects as well as any rccirculation conditions that may occur withbase relief geometries.

Adaptive methods are not new in numerical analysis. The first adaptive schemes weredesigned to compute automatically the time step in methods for the numerical solution ofordinary diffcrential equations. There the idea was to use a predictor-corrector algorithm witha higher-ordcr predictor, and to choose the time step to maintain accuracy and stability asdetermined by a rough error estimate or by computing an estimate of the truncation error. Theuse of adaptive mcthods in complex two-dimensional flow problems probably originated in theRussian literaturc in the 1981 monograph of N. N. Yanenko [21]. In Section 3.5 of that work,one can find a crude moving-mesh finite difference scheme for transonic flow around bluntbodies. While the numerical results obtained are not excellent, they nevertheless illustrate thepotential of such adaptive schemes.

Modern intercst in adaptive finite element methods was stimulated by the work of Babuskaand Rheinboldt and their co-workers [e.g. 22]. This work is largely confined to lincar ellipticproblems in onc and two dimensions and to linear parabolic problems in one space variable.Important contributions to the subject of adaptive schemes combined with multigrid methodsfor elliptic problems were made by Bank and Sherman [23] and Bank [24].

The use of adaptive schemes in space and time for parabolic problems has been discussed byFlaherty and his co-workers [e.g. 25]. A variety of moving finite element schemes have beenproposed, and we mention in this regard the works of Miller et al. [26], Diaz et al. [27], andDemkowicz et al. [28]. Recently, adaptive /Z-methods and moving mesh methods have alsobeen studied for two-dimensional problems in compressible flow (see, for example, the work ofDemkowicz and Oden [29], Oden and collaborators [3D] and Lohner and Morgan [31D.Adaptive papcrs employing finite difference approximations have also appeared, and wemention the survey articlc of Anderson [32]. However, these works describe adaptive schemesfor improving the structure of a difference grid and not necessarily for reducing local error. Theadaptive schemes of Berger and Jameson [33] and Berger and Oliger [34], on the other hand,use truncation errors in difference schemes as a basis for mesh refinement. Many results onadaptive finite elements can be found in the volume edited by Babuska et al. [35]. Acomprehensive critique and survey of the literature on adaptive finite element methods hasrecently been contributed by Oden and Demkowicz [36J.

Following this introduction, we review the equations, in conservation form, governingchemically reacting two-dimensional flow, with emphasis on the physics and chcmistry typicalof chemical lascrs. This is followed by a discussion of weak formulations and algorithms forexplicit solution of these equations on unstructured meshes. A brief discussion of error

,.

Chemically-reacting radiative flows 961

estimation is then presented, followed by an outline of an adaptive II-refinement algorithm.The results of scvcral numerical experiments are discussed in the final section and someconcluding comments are collected at the end of the paper.

2. MATHEMATICAL MODEL OF REACTING FLOWS

This section summarizes the basic partial differential equations and thermodynamic relationsused to describe the planar tlow of a chemically reacting. radiative, viscous, gas mixture. Theflow is composed of " ideal-gas species. m of which may represent vibrational levels of thelasing molecules (i.e. each vibrational level is treated as a separate species). Fluid motion isassumed to occur in the X - Y plane with X measuring distances along the primary flowdirection and Y the transverse coordinate or optical axis.

Among the equations used to describe a two-dimensional chemically-reacting flow are theto: conservation of mass, momentum, energy, species conservation. and the radiative transport

equations. Denoting spatial and time derivatives by ( )x' ( )y' and ( )" the conservation andradiation equations can be symbolically written as

(2.1)

with S. C and R representing the viscous, chemistry, and radiative effects, respectively. Each ofthese vector quantities contains 4 + " + m components which are given explicitly as

P pu pvpu pu2 +P puvpv

E=puv

F= pv2 + PII=<lpe + p)upe (pe + p)v

Pk PkU PkV

I,. 0 0-0

0 0(Ju, .•+ ax)'.,\'

0 0(Jxy.x + 0yy.)'

0 0S -~ 2 2 C= R=- 2: 2: (a ..u) . Qc QRI} } ./i=1 j=1

pi pr00 0 a" (T" - G)lv

• .. Here p is the total mass density. u and v are the velocity components, p is the tluid pressure, 0ij

are the components of the viscous stresses, e is the total energy defined by e = e + 1/2(u2 + v2)

where e is the thermodynamic internal energy per unit mass, Pk is the partial mass density ofspecies k, Oc and OR are the chemical and radiative volumetric heat source terms, pi and pfare the chemical and radiative mass exchange terms, I" is the average total radiation intensityof spectral transition (v + 1)- (v), a•. is a scale factor, Tv is the integrated gain, and G is thegain threshold.

The constitutivc relation for the viscous stress 0ij is given by

(J.. = Iilu .. + u.. J + AUk k{)"I} t I} }./ " I}

where Ii and A are the first and second coefficients of viscosity, Ui.j are the components of thevelocity gradient, and {)ij is the Kronecker delta. (For the sample problems of Section 4, theStokes relation A= -2/3µ has been assumed. which makes the bulk viscosity zero.)

In addition to the partial differential equation above, two thermodynamic relations are alsoneeded to close the system of equations. These relations are the ideal-gas state equation

p = (y-1)pe (2.2)


and an equation which relates the temperature to the internal energy

e = C" T.

Here y is the specific heat ratio and Cv is the mixture specific heat at constant volume. Withthese two additional equations we now have a complete system which can be solved for thevector of unknown quantities (p, u, v, e, Pk' It" p, T). once the chemical and radiative termshave been defined.

Chemistry termsWe emphasize that the algorithms and computational procedure discussed later in this paper

are applicable to quite general chemically-reacting flows with very complex flow domaingeometrics. For definiteness, in the present discussion we focus on chemical reactions occurringin a continuous-wave chemical laser cavity. Such reactions generally include a pumpingreaction and several deactivation reactions. The pumping reaction promotes molecules intoelevated vibrational and rotational quantum states which are subsequently deactivated byvibration-vibration (V-V) and vibration-translation (V - T) collisions and radiation. Otherreactions such as dissociation reactions are also known to occur in the mixing cavity butgenerally progress at much slower rates or have a significantly smaller impact on the lasingcalculations than the pumping and deactivation processes.

The entire system of reactions considered herein may be symbolically expressed as

where Xk represents one mole of species k. fl is the number of species, akr and bkr aredimensionless stoechiometric coefficients for the rth reaction, and krr and kl1r are the chemicalrate coefficients which define the rates at which the forward and backward reactions take place.These rates are assumed to be exprcssible in a generalized Arrhenius form

krr = Arr TO" exp( - Err/T)kl1r = Al1r TO., exp( - Ebr/T)

where Arn Abn Bbn Ern and El1r are reaction-dependent constants and T is the temperature (inKelvin).

For each of the chemical reactions taking place in laser cavity a production rate wr may becalculated which represents the rate of progress of the rth reaction. A convenient expressionfor this rate is given by

n 11

wr = krr IT (Pj/ Mj)"k<- kbr IT (p) Mjtbj=1 j=1

where II represents the pi product over all species taking place in reaction r, and Mj is themolecular weight to species j. With this definition of the production rate, the chemical sourceterms in the species conservation equations are

pi = Mj 2: (bj, - Uj,)w,,and the chemical source term in the internal energy equation is

where q, is the heat of reaction for the rth reaction at 0 K.

Radiation termsThe radiation transport equations given in (2.lf) are essentially the transient form of the

radiation transport-theory description of a chemical laser outlincd by Emanuel [1] and

r

r.


. .

. -

extended by Ramshaw in [20]. The principal assumptions upon which this description is basedare: (a) the optical properties of the lasing cavity are those of a Fabry-Perot cavity(plane-parallel mirrors); (b) each vibrational level is represented as a separate species which isin rotational equilibrium; (c) Doppler line broadening is the only significant line-broadeningmechanism: (d) only a single P-branch transition [(v + 1, I - 1)~ (v, I)] can lase for eachvibrational transition; and (e) steady-state lasing conditions are unique and independent of thetransient history.

The influence of the radiation intensity on the fluid dynamic equations occurs through thesource terms which must be added to the species equations and to the energy equation. Theseterms are given by

with the convention that I" = 0 for v = - 1 and v equals the highest vibrational level underconsideration, and pf = 0 for every species that is not one of the vibrational levels of the lasingspecies. In these expressions. h is Planck's constant. k is Boltzman's constant. NA is Avagadro'snumber, Mj is the molecular weight of species j, 0-., is the P-branch gain per unit length. Vl' isthe P-branch spectral frequency, I~ is the rotational level of largest gain for the vibrationaltransition (v + 1)~ (v), and Or is the rotational temperature of the lasing species defined byOr = hcB,.Ik, where c is the speed of light and Bu isa vibrational constant.

Based upon the assumptions listed above. the P-branch gain per unit length in an activemedium may be expressed as

NAhv(v.J) (21 + 1 )a-(v. J) = "I. 21 _ I PU+!.J-I - P".J B(v. J)CPo(v, I»

J

where Pv,l is the mass density of the lasing species in level (v, I), B(v. I) is the P-branchEinstein coefficient, and CPo is the Doppler line shape factor evaluated at line center. Thesequantities are given by

_ {Or(21 + 1) [-J(J+l)Or]}Pu.l - Pu T exp T

( C )( M- )112CPo(v(v. I» = v(v,J) 21£kt!i" T

81£3 . I )B(v. J) = 3h2c (21 + 1 IM'oJI2

where IM"ll is the irreducible part of the dipole matrix for the P-branch transition(v + 1. J - 1)~ (v, I). The expression used to evaluate the dipole matrix element is

where aM and bM are constants depending on the type of gas laser.The final quantity needed to evaluate the gain per unit length is the P-branch spectral

frequency v(v, J). The form of the equation used to evaluate the spectral frequency is given by

where WC' WeXe, WeYe and Bc are conventional spectroscopic constants.

Weak formsIt is well known that in actual chemically-reacting flows, the governing eqns (2.1) are either

inappropriate or inadequate for defining an acceptable model: they do not include jump


conditions across shocks and contact discontinuities (for large and zero Reynold's numbers),they do not include boundary or inertial conditions, and they presume an unrealistic degree ofsmoothness of the solution vector U. These shortcomings are overcome by reformulating theproblem in a weak (distributional or variational) sense which is fully equivalent to (2.1)whenever solutions U are sufficiently smooth.

In developing a mathematical formulation of general chemical laser problems, several basicconcepts are kept in mind. First, it must be recognized that the conservation and balance lawsof mechanics and physics are global laws which apply to finite volumes of material. They arethus given to us in an integral form, and it is legitimate to extract from them systems of partialdifferential equations, as is done in most fluid mechanics studies only when the motions andfluxes are sufficiently smooth. Secondly, in many classes of flow problems, particularly those ofpure hyperbolic character or those involving significant convective terms, the time variable t isintrinsically connected with spatial independent variables Xi' i= 1, 2, 3 through kinematics andthrough physics (as, for example, in jump conditions characterizing, conservation laws acrosssurfaces of discontinuity).

For the class of problems considered here, a weak formulation is defined in terms of twoclasses of functions: V, the class of trial functions, to which the solution U belongs, and W, theclass of test (or weight) functions which are integrated against the residual of the governingequations of chemically reactive tlows. The resulting weak form is:

find U in a class V such that

T T T

i( (U;4j»-ET4j»x-FT4j»y)dQdt=i r (S+C+R)T4j»dQdt+i ( rT4j»dsdt (2.3)

o JQ II Jo 0 lcmfor all test functions 4j» = {<p., <Pz, .... <P4+/I +m} in W, where [0, TJ is the time interval ofinterest, Q is the region through which the fluid moves, BQ is the boundary of the flow regionQ, and r is the vector of boundary fluxes. It is understood that the viscous stress terms on theright-hand side of (2.3) may also appear in the integrated form,

so that differentiability of oij in L 1 is not necessarily required.Our numerical approximation of the laser problem will begin with a discrete approximation

of a weak form of the problem such as that in (2.3).

3. NUMERICAL MODELS AND ALGORITHMS

We now discuss a two-step semi-explicit. finite element based, Lax- Wendroff/Taylor-Galerkin scheme used to advance the solution of (2.1) in time. The method is not particularly I ,

robust, and can produce spurious hourglass oscillations near shocks; but these can besuppressed using artificial viscosity and FCT (flux-corrected-transport) schemes. We have alsocoded other algorithms for adaptive schemes, and the algorithms used in this report are not tobe regarded as our final choices for general adaptive approaches for lasing simulations. Thelocal error estimation procedures and adaptive mesh strategy is also outlined below.Concluding this section is a brief description of the technique used to incorporate the chemistryand radiation effects.

Basic discrete formulationWe now fashion from (2.3) a specific algorithm for modeling supersonic chemically reactive

flow of a viscous compressible gas. Let the flux Q, the matrix T be defined as

Chemically-reacting radiative !lows

pu pvpU2 + p(U) PUVPUV pV2 + p(U)(pe + p(U))U (pe + p(U»v

Q =1 ~IUP1V

PllU PllV

0 0

.0 0

0 0-all -Ot2

T-ra

"

-022

.. ~OllU - 012V -012U - °22V

0 0

965

(3.1)

for all cpE W(3.2)

• r

.-

each having 4 + 11 + m components and where p(U) is the pressure given by eqn (2.2). Theparticular weak form of the governing equations follows from (2.3) for a time interval [t'l' t'2]as:

Find U = U(x. t) E V such that

_JfZ r UT cp,dO dt + r U'f'(x. t'2)CP(X.t'2) dO -1 UT(x, t'.)CP(X. t'1) dOf, JO(,) JO(f2) O(fl)= Jf' r Q(U): V'cpdO dt - If' 1 cpT(Q(U)n ds dl

f, JO(,) " ao(,)

+ Jf' r T(U): V'cp+ (CT + R1)cp dO dt - Jf' 1 cpT(T(U))n ds dl'I JO(,) f, aQ(,)

Here, cp= {4>I' 4>2,.... 4>4+lI+m}T. dO = dx. 4>",= 4>",(x.t). x E O(t). a' = 1, 2 ..... 4 + 11 + m,cp,= acp/ at. and n is an outward unit normal vector to the boundary.

It is easily verified that, when Q and T are given by (3.1), (3.2) is equivalent to the entiresystem of Navier-Stokes equations, Rankille-Hugoniot jump conditions. and initial conditionson U (at t = t'1) whenever U is a Cl-function everywhere except at surfaces of discontinuitywhere the jump conditions hold.

In a strictly formal way, the finite element approximation of the flow is obtained from theweak statement of the conservation laws. by interpreting 0 as a quadrilateral element,replacing U by the discrete approximation Uh and replacing the test functions cpby the discretetest functions cph. Thus, over each element Oe(t), e = 1. 2 .... , E we must have:

-I'zl UhTcp~'dOdl+ L Uh(x.'t2)Tcph{x.t'2)dQ-l Uh(X,t'lfcph(X.t'l)dQ" 0,(,) Q,( 'z) O•.(T,)

= I"l Q(Uh): V'cphdQ dt - I"L cp1:T(Q(Uh))n ds dtf, Q,(,) 1, ao(,)

+ I" L T(uh): V'cph+ (ChT + Rhr)cph dQ dtTl 0,(')

- ('2 L cphTT(Uh)n ds dt for all test functions cph in W"JT, ao(,)(3.3)


Introducing the local approximation of Uh and choosing an appropriate numerical integrationrule to evaluate the time integral yields a local finite element model of the conservation laws.

A Taylor-Galerkin/Lax-Wendroff scheme is derived from (3.3) by using the midpointintegration rule:

Partitioning the time interval of interest into a finite number of steps,

O=tO<t1<tc<" ·<tN=T

one arrives at the following two-step scheme:First step. For each element Qe, calculate a constant element vector U~~1I2 from

4 {(1 ).!:!.t A"+1 (1 a· }A"+IUTI+l12= ~ 1/.dQ U',T1 __ ~ tJ1'dQ)Q;.lIe a.e L.J '#', a 2 ATI+ 112 ~ Q'.f3

;=1 ~2~ e Q~+ll2 oXp

where A; is the area of the element at time t", U~" is the a-component of vector U at time tTl atnode i, tJ1; is a piecewise bilinear shape function which has a value of unity at node i and is zeroat every other node, and a = 1,2 .. , .. 4+ n.

Second step. For each node, calculate U~"+1 by solving the following system of equations:

f (r tJ1;tJ1jdQ)U~II+1 = f (r tJ1;tJ1jdQ)U~T1·+1j=1 JO"'l j=l Jo"

+!:!.t r Q~pll2 al/J; dQJO"+1 axp

+!:!.t r tJ1;(C~ + R~) dQ + dt r r:.p atJ1; dQJon Jon aXil

-!:!.t 1 nf3(Q':,.P112 - Q':,.p)l/J; dS -!:!.t 1 nfJQ~ptJ1; dSann+1 30"+1

-!:!.t 1 nf3(T~pll2 - T~p)tpi dS -!:!.t 1 np T':,.f3tJ1; dSaon-l aon+\

Here, Q:p denotes an elementwise averaged value of the flux, and N is the total number ofnodes in the discretization.

Error estimationWe shall discuss here some general properties of the evolution and distribution of error in

finite element meshes and the use of so-called interpolation error estimates. These easilyimplementable local estimators give a correct indication of relative error between successivemeshes or approximation orders and, thus, correctly direct an adaptive strategy to systemati-cally reduce local error.

Interpolation errors. Let u be a smooth function defined over a regular domain Q. TheWr'P(Q)-semi-norm of u is defined by

{L [ ai+iu]p } liplulnn.p(Q) = .2: ~ .i ~ j dQ

O,+/=r vXIVX2;.j~O

where 1 ~ P < 00 and r is a non-negative integer. For the special case of p = 00, which is also ofinterest, the Wr'P(Q)-semi-norm is given by

Iai+iu(x) I

lullY">(O) = max milx . .nO '-:l=r ax; ax~

I./~O

With these definitions of the semi-norm, the Sobolev norm of u is then

{

r } lipIlulllY"p(o) = t;o lul~,.p(o)


Let G be an arbitrary convex subdomain (a finite element) of Q over which u is interpolatedby a function iii, which contains complete piecewise polynomials of degree k. Then, it can beshown [36] that the local interpolation error in the W'·P(Q)-semi-norm is

where h = the diameter of the domain g, y = the diameter of the largest sphere that can beinscribed inside G, n = the dimension of the domain Q, p . q = real numbers, 1:!!ii;. p. q:!!ii;.00, andC = a constant independent of h, y, and u. If y is proportional to h, and if it remainsproportional in refinements of G defined by parameterically reducing h. we have

/E"I :!!ii;.Ch"lq-lIlp+k+I-", lui""q.G k+l.p

with I . Im.q.G = I . Iwm.q(G)' etc. and Eh = u - ah'By the triangle inequality, if u" is the finite element approximation of u, then

(3.4)

lu - uhl",.q,G:!!ii;.lu -llhlm.q.G + lah - uhlm.q.G

The problem then reduces to one of estimating

la" - uhlm.q.G

i.e. the semi-norm of the difference between the interpolant iih of the exact solution and thefinite element aprpoximation Uh' This is problem-dependent. Strouboulis and Oden [37] haveshown that for the linear elliptic problem - e Li.u + pu = J,

h2

lIah - uhIlWI.a(o,):!!ii;.llu - ahlll.".o, + 2: C yL k(h) lulvc.o,I. L

for linear shape functions, where k(h) is a term dependent on properties of Green's functionsfor the operator and on FEM approximations of Green's functions.

Due to special superconvergence properties of finite element methods, the correction term,lall - uIII""q.n, can be several orders of magnitude smaller than the interpolation term forconcrete cases [37]. For this reason, we choose as error indicators in the present paper

B:,·q = IIEhllm.q.G (G = Q..)

Some choices for n, m, k, p and q of interest are:

(i) n = 2. m = 0, k = 1. p = q = 2, then

Be :!!ii;.Ch21u 12.2.n,; 0.. = 0~·2

'.. In this case, one must approximate the W2•2 semi-norm of u over Q ..; i.e. the L2-norm of

second partial derivatives of u. The error indicator 8.. is then set equal to IEhIL2(o,) for finiteelement Q ...

(ii) n = 2, p = 00, q = 1, k = 0, m = 0, then

B.. = Ch21Elliuverage:!!ii;.Ch3 lu II. ... G

:!!ii;.eh3 max IV· u(x)l.xeG

Then we have for the error indicator 8e•

In all of these cases. it is also possible to estimate the constant C. We shall not describe howthis is done in the present study.


Adaptive mesh strategyWe outline here the basic adaptive strategy. The method used was developed and used in

[301; it suffices to provide only a brief outline.Suppose that an error indicator Be can be calculated for each finite element Qe in a given

mesh at time t. The error indicator is, in general, a real number representing the local error ina suitable norm, and it is computed using one of the procedures described in the precedingsection. The decision to refine the mesh is based on whether or not local error indicatorsexceed preassigned tolerances. We shall describe the adaptive procedure used here.

An h-rejinement/unrejinement method. Our II-procedure involves the following steps:(1) For a given domain Q, such as that shown in Fig. 1, a coarse finite element mesh is

constructed which contains only a number of elements sufficient to model basic geometricalfeatures of the flow domain.

(2) As our adaptive process will be designed to handle groups of four elements at a time. wemay generate a finer starting grid by a bisection process, indicated in Fig. la, to obtain aninitial set of element groups.

(3) We initiate the numerical solution procedures on this initial coarse grid, and computeerror indicators Be over all M elements in the grid. Let

(4) Next. we scan groups of a fixed number P of elcments computeP

(J~ROUP = L Be.k=1

where ek is the element for group k. We take P = 4 in our current code.(5) Error tolcrances are defined by two real numbers. 0 < ll', {3 < 1. If

(Je ;:;::(3BMAX

(a)

\

.f

;,

<;

Refine

iJnrefme

(b)

Fig. I. (a) A coarse initial mesh consisting of four-clement groups and (b) refinement andunrefinemenl of a four-element group.

Chemically-reacting radiative flows

we refine element Be. This is done by bisecting fJe into four new subelements. If

etROUI' "'" /YBMAX

969

we unrefine group k by replacing this group with a single new element with nodes coincidentwith the corner nodes of the group.

This general process can be followed for any choice of an error indicator.As an example of our mesh refinement strategy, consider the uniform grid of four elements

shown in Fig. 2a and suppose that the error estimators dictate that element A is to be refined.Thus, A is divided into four elements, L 2, 3, 4, and shown, and the solution values at thejunction nodes, shown circled in the figure, are constrained to coincide with the averagedvalues between those marked X.

Next, assume that an additional refinement is required, and that we must reflne element 3.We impose the restriction that each element side can have no more than two elementsconnected to it. Thus. before 3 can be refined, element B must first be reflned. The constrainednode Bl in Fig. 2b now becomes active, while node Cl remains a constrained node. With Bbisected, we proceed to refine 3 into sub-elements /Y, ~, y, D and new constrained nodes, againcircled in Fig. 2d, are produced. In this case, only element B had to be refined first in order to

." refine 3, but, in general. the number of elements that must be refined in order to refine aparticular element cannot be specified.

Radiation and chemical controlThe actual procedure for solving the partial differential equations outlined in Section 2

involves an operator splitting technique similar to that employed in [17]. In the approachadopted here, one first solves the continuity, momentum energy, and species equations usingthe Lax-Wendroff/Taylor-Galerkin scheme for a first approximate solution UA+1 at timetn + /1e, This solution, U~~+I,is obtained with only the viscous contributions included on theright hand side of eqn (2.1).

The next step in the computation is the chemistry calculation. The chemistry changes to the

A I BI

~

I=>.. (a 1 t'i..

4 3 8 7

1 2 5 6

4 3BI B

2 1

C1

C D

I b 1

cc) ldl

x -ACTIVE NODE

o -CONSTRAINED NODE

Fig. 2. Sequence of refinements of s unifrom mesh.


(4.1)

energy and species are computed by solving

ape-=LQ(Opt ,"

and

for a second approximation of the solution UB+1. Since these equations do not include spatial

derivatives. the updating of the species and energy values is performed in a local manner ateach nodal location. The integration method used here is a simple forward Euler integrationwith the total time interval /),.t divided into a minimum of four substeps. More steps than thisare sometimes used depending on the magnitude of the production rate w,.

The final step in the computation is the radiative calculation. which again involves modifyingthe species and energy values. The first step in the radiative calculation is to update theradiation intensities l~:to I~+I. These intensities are functions only of the primary flow directionX and time. and evolve according to eqn (2.1f). With I~+I known. the radiation source termscan be evaluated and the energy and species updated by solving

ape = 2k(), L J~I"et"at It v Vv

andapk = Mk (l,.etv _ Iv-tetv-I)at NAil Vv Vv_1

for the final values of field variables at time tn + /),.t. The integration of these equations is alsoperformed using a forward explicit method with the integration carried out at the nodallocations.

4. SAMPLE RESULTS

We present here some preliminary numerical results for the steady-state lasing conditions inthe cavity and base region of a continuous-wave chemical laser. The basic flowfield geometry isshown in Figs 3 and 4. Our aim is to present a wide variety of results for a number oftwo-dimensional problems with different inflow conditions and nozzle configurations which arerepresentative to typical gas lasers. The problems described here are not for any particular typeof gas laser. such as a HF or DF laser, but include all the key ingredients found in suchchemically reactive flow regimes. The problems have been selected to demonstrate theversatility and ease with which our adaptive strategy copes with the changing conditions.

Two commonly encountered nozzle configurations are included in the sample calculations.These include configurations with and without base relief as shown in Fig. 4. Thecomputational regions for the first three sample problems are also outlined in this figure wheresymmetry between adjacent nozzles has been assumed.

A simplified chemical model involving five species and three chemical reactions is employedin this investigation. These reactions include a pumping reaction and a deactivation reactionspecified by the following balance equations:

A+B~D(u), u=O.l

C + D(l)~C + D(O)

Reaction (4.1a) represents the pumping reaction which creates vibrationally excited productspecies that are subsequently deactivated by the vibration-translation reaction (4.1b) and byradiation. The rate coefficients and heats of reaction for these reactions arc given in Table 1which includes values similar to those in [2] for the HF lasing system. The molecular weightsfor the species and the specific heats at constant volume are given in Table 2.


::Outpm Beam

z

l:::v

~ "",'A,;,

~Row Axi~

...

. .

Fig. 3. Cavity schematic and coordinate system.

1------'1 Computotlonol II reqlon I------ .....

Oxidizer

• Fuel

{ol

• Oxidizer

• Fuel

{b}

Fig. 4. Typical nozzle configuration (a) without base relief and (b) with base relief.


Tahlc 1. Chemical ralc coefficients and heats of reaction. (K =A,B exp(-E/T) ("n3/mole_s): all isin g-cm/mole-s2

)

Forward Backward-MI

Reaction A" 8" E" Ah' 11lH Eb. (II)

A+ B¢>D(O) 9.0 x 1012 0 800 3.57 X 10.2 0.15 5700 4.134 x 10"A+8¢>D(I) 1.8 x 1013 Il 800 9.0 x 1012 0.15 16700 2.692 x Ill"C + D(I)¢>C + D(O) 3.0 x 10" -1.43 0 3.0 x 10'6 -1.43 5180 0

Tablc 2. Species molecular weights andspccific hcats at constaot volumc (N-

cm/g·K)

Specics A B C D

Mol. WI 21.8 4.3 15 26.1C.. 560 170 120 200

A number of other parameters is also needed to start the lasing calculations. These includethe dynamic viscosity, the spectrosonic constants. the dipole matrix constants, and therotational temperature constants. A list of the values used in this work is given in Table 3.

The implementation of the adaptive procedure requires the initialization of two thresholdrefinement parameters (l' and p. a maximum level of refinement parameter, and an errorindicator flag. The threshold values were taken to be -0.1 and 0.15. respectively, whichspecifies that all elements with an error indicator greater than 15% of the maximum errorindicator will be refincd and no unrefinemcnt will take place. (The specification of nounrefinement was used simply to save computer run time.) Thc maximum Icvel of refinementwas set to 3, which indicates that each initial element in the mcsh may be subdivided into amaximum of 64 smaller clements. The error indicator flag. used to set which field variable willbe used in computing the error indicators, is set to the total density in the first two sampleproblems while other quantities have been considered in sample problems 3 and 4.

The numerical examples have been subdivided into four groups. In the first two groups weconsider nozzle arrays with and without base relief as shown in Fig. 4. In these problems. arectangular portion of the lasing cavity has been taken as the computational region and thetotal density has been used to compute the error indicators. In the the third and fourth groupsof problems, some results are presented using other field variables for calculating the errorindicator, and in sample problem 4. results are given for a somewhat more complex geometry.We believe that these results represent the first dynamically adaptive mesh refinement methodsfor supersonic flow lasing calculations.

The numerical results presented below include plots of the final adaptive meshes. velocityvector plots and contour plots of several of the field variables, including the total mass density.pressure, Mach number. and species densities. The contour plots have minimum and maximum

Table 3. Flow field and lasingparameters

Dynamic visco~il~ (g/cm·s)µ=2.678xlOSpectroscopic constants (cm - 1)We =4138.5wet'e = 0.0988..=11.007Mirror reUectivilies" =0.98'2 = 0.98Dipole matrix constantsa", = 0.70b", =0.03Rotational temperature constants8,. = 11.007 - 0.293(v + 1)

...,

- .


Oxidizer nozzle centerline I no- flow boundory

0.50 cm

Fuel noule centerUne I no - flo,*" t)oundory

973

Fig. 5. Schematic diagram of computational region without base relief.

contour line values given above the figure, with high and low contour regions marked with the•• letters Hand L. (The units used in these calculations are cgs unless otherwise noted.).'

Sample problem 1In our first problem, we consider the supersonic flow of an oxidizer and fuel stream through

a rectangular portion of the lasing cavity just downstream of a nozzle array constructed withoutbase relief (see Fig. 4a). Each of the inflow streams entering the computational region isassumed to have uniform transverse flow properties with an average of these values taken atthe intermediate point bctween the two streams.

The computational region, shown enlarged in Fig. 5, is composed of two inflow boundarieson the left, where the fuel and oxidizer streams enter, an outflow boundary on the right, andtwo no-flow boundaries on the top and bottom edges which are assumed to be symmetry planesin the laser cavity. This region is initially divided into fifty rectangular bilinear elements withfour elcments spanning the oxidizer inflow region. one element spanning the fuel inflow planeand ten downstream elemcnts. While this initial discretization is quite crude, it captures all ofthe necessary detail to implement the adaptive strategy and automatically refine the grid up tothe maximum level specified. By prescribing the maximum level to be three, we have thuslimited this example to a maximum of 3200 elements with all subsequent refinements based onan error indicator calculated from the total density.

It is well established in the literature that large variations in the inflow conditions may existfor different nozzle shapcs and lasing gases. We have therefore considered three differentcombinations of inflow conditions which appear to be consistent with sampic data reportedelsewhere. The prescribed data at the inflow plane for the primary and secondary nozzles isgiven in Table 4. With this data all other field variables necessary to describe the inflow

.' Tablc 4. Prcscribed inflow conditions for oxidizer and fuel strcams

Mass fractions for speciesT Pressurc v-velocity

(K) Mach No. (N/cm2) (cm/s) Cp/C,. A B C

SellOxidizer 500 2.0 0.2 0.0 1.4 0.0 0.2 O.llFucl 110 1.5 0.15 0.0 1.4 0.2 0.0 0.8Inlerface - - - - - 0.1 0.\ 0.8

Set 2Oxidizer 500 2.0 0.4 0.0 1.4 0.0 ll.2 0.8Fucl 400 1.5 0.3 0.0 1.4 0.2 0.0 0.8Interface - - - - - 0.\ 0.\ 0.8

Set 3Oxidizer 400 3.0 0.4 0.0 1.4 0.0 0.2 0.8Fuel ItO 2.0 0.15 0.0 1.4 0.2 0.0 0.8Interface - - - - - 0.1 0.1 0.8


+

Conrour Line Data: mln= 4.50 E·6 max= 7.50 E·6 inlerval= 0.5 E·6 (glcm3)

Fig. 6. Adaptive mesh and total mass density conlours for data set one. an inviscid solution with nobase relief.

-.

,..'.

conditions are calculated to satisfy the thermodynamic constraints. Initial conditions on thedownstream variables have been set equal inflow values with the exception that downstream ofthe intermediate point between the two nozzles with flow was initially assumed to consist onlyof species c.

For the first group of data listed in Table 4. an adaptive solution was calculated for threedifferent cases. including an inviscid case without chemistry. a viscous case without chemistry.and a viscous case with chemistry and lasing. The resulting mesh for each of these calculationsis shown in Figs 6-8. Comparing these grids. we see that the inviscid and lasing calculationsproduced a similar resulting mesh with the purely viscous calculations concentrating therefinement more toward the upstream interface region.

The plots of the density contours show a similar trend for all three cases with the smallerdensity values occurring around the inflow region of the fuel stream and the higher densitycontours occurring in the upper left portion of the flow region and downstream of the oxidizertlow. The inviscid and lasing density contours also show a density trough extending diagonallyacross the computational region which has been reflected in their final adaptive meshes.

For the lasing calculations. plots of the Mach number contours. pressure contours, and thespecies density contours for species D(O) and D(1) were also produced (see Figs 9 and 10). TheMach number contours are fairly uniform with values ranging from 1.29 in the downstreamregion of the fuel nozzle to 2.17 in the downstream region from the oxidizer nozzle flow.

The second example dealing with a nozzle array without base relief involves a perturbationof the first group of data whereby the inflow temperature in the fuel stream has been increasedto 400 K and the inflow pressures in both streams have been doubled (see Table 4). For thisprescribed data, we have computed a solution for the viscous case without chemistry and forthe viscous lasing case. The resulting meshes shown in Figs ]] and 12 are similar in characterwith considerable detailed refinement now occurring along the diagonal of the computationalregion. A plot of the density contours for these two cases shows a density trough extendingdiagonally across both plots as was present in the density contours from the first set of lasing

, ,

.'.<

..


-f

Conlour Line Data: min= 4.50 E-6 max= 7.50 E·6 inlerval= 0.5 E·6

Fig. 7. Adaptive mesh and total mass density contours for data set one. a no base relief viscoussolution without chemistry.

H-

Conlour Line Data: min= 4.50 E·6 max= 8.50 E·6 inlerval= 0.5 E·6

Fig. 8. Adaptive mesh and total mass density contours for data set one. a viscous lasing solution withno base relief.

975

:sa-.

Contour Line Data: min= 1.30 max= 2.10 inlerval= 0.1 Contour Line Data: min= 0.10 E-? max= 2.10 E-? interval= 0.4 E-? (g/cm3)

~

:-loornz

:!::c:l:>V>Vl

'"::li:l.....

(g/cm3)

"

(a I

Contour Line Da13: min= 0.20 E·7 max= 1.40 E-7 interval= 0.2 E-?

c::==:>

.~><===-

interval= 0.0 I {N/cm21

<>

••• Q

~~C:D 0 <:::>..!1-~

(a)

max: 0.20

6

~

«/

<::::l

Contour Line Da13: min: 0.15

( bl

Fig. 9. Contour plot of (a) Mach numbers and (b) pressures for data set one. aviscous lasing solution without base relief.

Fig. 10.

(b)

Species contour plots for species (a) D(O) and (b) D(l) for data setone_ a viscous lasing solution without base relief.

." "


Contour Line Data: min= 2.30 E·5 max= 3.40 E·5 interval= 0.1 E·5

Fig. 11. Adaptive mcsh and total mass density contours for data sct two, a no base rclief viscollssolution without chemistry.

i ~,

++:±

Contour Line Data: min= 2.10 E·5 max= 3.50 E-5 inlerva1= 0.2 E·5

Fig. 12. Adaptive mesh and total mass density COnlours for data set two. a viscous lasing solution withno base relief.

977

978 J. M. BASS and .I. T. ODEN

Contour Line Data: min= 0.30 max= 0.50 interval= 0.03 (Nfcm2)

Fig. 13. Pressure contours for data set two. a viscous lasing solution without base relief.

calculations. The significant difference between these density contours and those of theprevious example is that the high denisty contours have now shifted to a region downstream of ~.fuel inflow.

A plot of the pressure contours and species densities for species D(O) and D(l) is shown inFigs 13 and 14. The pressure contours show a similar pressure trough as in the first group ofinflow data with a peak pressure of 0.509, now occurring at a point directly downstream fromthe nozzle interface.

Contour Line Data: min" 0.20 E·6 max" 2.60 E·6 interval" 0.4 E-6

~-(a)

Contour Line Data: min" 1.40 E·g max" 9.40 E·g inlerval= 2.0 E·g

lb)

Fig. 14. Species contour plots for species (a) D(O) and (b) D(1) for data set two. a viscous lasingsolution without base relief.


Fig. 15. Adaptive mesh for data set three, a viscous lasing solution without base relief.

979

The final problem selected for study with this geometry is also a perturbation of the firstexample where the Mach numbers in the oxidizer and fuel streams have been increased to 3.0

,. and 2.0. respectively. and the pressure in the oxidizer stream has been doubled (see Table 4).The resulting mesh for the lasing calculation is shown in Fig. 15. Here. we see almost a uniformfine refinement of the lower diagonal half of the computational region. The density andpressure contours (Fig. 16) show the troughs that previously extended across the diagonal ofthe computational region have now pushed downward toward the nozzle interface with similar

Contour Line Data: min= 1.10 E·S max= 3.50 E·S intervale OJ E.S

(a)

->

Contour Line Dal.1: min= .12 max: 0.48 inlerval= 0.6 (N/cm2)

(b)

Fig. 16. Contour plot of (a) total mass density and (b) pressures for data set three. a viscous lasingsolution without base relief.


Conlour Line Data: min= 0.20 E·6 max= 1.70 E-6 imerval= 0.3 E·6

(a)

Contour Line Data: min= 0.10 E·7 max= 1.60 E-7 imerval= 0.3 E· 7

...

"-"

(b)

Fig. 17. Species contour plots for species (a) D(O) and (b) D(1) for data set lhree. ,I viscous lasingsolution without base relief.

peak values occurring as in example two. Plots of the species densities D(O) and D(l) arc alsoshown in Fig. 17.

Sample problem 2This problem examines the supersonic flow of an oxidizer and fuel stream through a

rectangular portion of the lasing cavity just downstream of a nozzle array constructed with baserelief (see Fig. 4b). The computational region. shown enlarged in Fig. 18, consists of two

Ol.ldlZe~ nozzle centerlIne/no - How boundary

0.35 em

EuIf)

Qo

EuIf)Noo

Fuel nOlzle centerlln.e / no - f\ow boundary

Fig. lIt Schematic diagram of computational region with base relief.

Chemically-reacting radiative flows. 981

...

inflow boundaries on the left separated by a no-flow region. an outtlow boundary on the right.and no-flow boundaries on the top and bottom. This region is initially subdivided into 98 finiteelements with four elements spanning the oxidizer inflow plane. two elements spanning the fuelinflow plane and fourteen downstream elements. Combining such an initial mesh with amaximum level of refinement of three, limits the number of elements for this problem toapprox. 6275.

In conjunction with this new geometry, the same three sets of inflow data as for the nozzlearray with no base relief are considered (see Table 4). The initial conditions for thedownstream variables have been set equal to the inflow values for every point. All subsequentmesh refinements obtained for this problem are based on error indicators calculated with thetotal mass density.

For the first set of data in Table 4. two adaptive solutions. including a viscous case withoutchemistry and a viscous lasing case. have been calculated. The resulting meshes (Figs 19 and20). show a similar trend of high refinement directly downstream of the base relief section. withthe purely viscous case localizing the refinement much further upstream. (This result for thepurely viscous case is similar to that obtained in the first sample problem for the same set ofinflow data, see Fig. 7.) The contour plots of the total density for these two cases are verysimilar with regions of low density occurring near the base relief section of the intlow planeand regions of high density occurring near the oxidizer inflow plane and downstream of theoxidizer intlow.

With the data obtained from the lasing calculations, contour plots of the pressure and speciesdensities D(O) and D(l) were produced (Figs 21 and 22). The pressure contours (Fig. 21) showthe same trends as the total density contours with low pressure values occurring near the baserelief section and a high pressure of 0.27 N/cm2 occurring near the inflow of the oxidizerstream.

Contour Line Data: min= 0.10 E·6 max= 8.] 0 E·6 interval= 0.5 E·6 (g/cm3)

Fig. 19. Adaptive mesh and total mass density contours for data set one, a viscous base relief solutionwithout chemistry.


I II I

I I II I I

i+

-I I I..,

I I I II I

Contour Line Data: min: 0.10 E·6 max: 8.60 E·6 interval= 0.5 E·6 (g/cm3)

Fig. 20. Adaptive mesh and total mass density contours for data set one. a viscous lasing solution withbase relief.

.~,

The contour plot of the D(l) species density (Fig. 22b), shows the region of highest densityoccurring almost directly downstream of the base relief section of the inflow plane. Themaximum density contour value in this plot is 4.5 X 10-7 g/cm3 which is approx. 3 times themaximum value obtained with the same data for the nozzle array without base relief.

With the second set of inflow conditions in Table 4, a lasing solution was calculated with theresulting adaptive mesh and total density contours given in Fig. 23. The adaptive meshobtained here is similar to the lasing results for the first set of inflow data. with the finerefinement extending the full length of computational region. The contour plot of the totaldensity, Fig. 23b. has a low density region near the base relief area. and high density contoursoccurring downstream of the fuel inflow. This is the same qualitative result obtained for the nobase relief solution (see Fig. 12).

Contour line dora: min; 0.07 max; 0.27

00o

Interval; 0.02 IN/cm21

Fig. 21. Pressure contours for data sel one, a viscous lasing solution with base relief.

~

(h)

Fig. 22. Species contour plots for species (a) D(O) and (b) D(l) for data setone. a viscous lasing solution with base relief.

(j:r

'"3,,'e:.f'".,~;;

00...OJc..21"<'t1l

::>o~II>

(g/cml)min= 0.40 E-5 max= 3.20 E-5 interval= 0.4 E-5

Fig. 23. Adaptive mesh and total mass density contours for data set two. aviscous lasing solution with base relief.

Contour Line Data:

(g/eml)

(g/cml)

~S>-

o

(a)

Conlour Line Data: min= 0.30 E-g max= 2.10 E-8 inrervaJ= 0.3 E·g

Conrour Line Data: min= l.()() E-? max= 4.50 E-? interval= 0.5 E-?

'"00'""


Maximum velocity: 9.85 E-I em/sec

----------------------

Fig. 24. Velocity vector plot for data set two. a viscous lasing solution with base relief.

From the lasing solution plots of the velocity vectors and species density contours for speciesD(O) and D(l) were extracted. The velocity vector plot, Fig. 24, depicts an almost stagnantregion neaf the base relief section of the inflow plane and a bending of both the oxidizer andfuel streams around the no-flow region. At the outflow plane on the right, a cross section of thevelocity vectors shows peak values of streamwise velocity occurring at the nozzle centerlines.with a somewhat smaller value in the mixing region. For these inflow conditions, a maximumvelocity of 9.85 x 104 cm/s occurred approximately on the nozzle centerline of the oxidizerstream. From this plot, it does not appear that a region of recirculation exists near the baserelief area.

Species contours for species D(O) and D(l) are shown in Fig. 25. The species D(O) density

Conlour Line Data: min" 0.20 E·6 mllJl= 3.20 E·6 interval= (l.5 E·6 (g/cm3)

-..

C> •

(a'Conlour Line Data: min= 0.30 E·7 max= 3.30 E·7 interval= 0.5 E· 7 (glem3)

-.

(b)

Fig. 25. Species contour rlols for species (a) 0(0) and (b) 0(1) for data set two. a viscous lasingsolution with base relief.


..

contours are significantly larger than the values obtained with the first set of inflow data and areapprox. 30% larger than the maximum contour values obtained for the same data without baserelief. The species D(l) contour values ranged from 0.3 to 3.3 x 10-7 g/cm3• which are slightlysmaller than the contours for the first data set, but were up to three times larger than theresults obtained from the case without base relief.

Using the third group of inflow data from Table 4, numerical solutions were obtained for apurely viscous case and a lasing case. The resulting adaptive meshes, which were nearlyidentical (see Figs 26 and 27), have a wide region of fine refinement extending the entire lengthof the computational domain. with a slight biasing of grid toward the downstream region of thefuel inflow. The total density contours are also quite similar, with the lasing solution having aslightly higher maximum value. As with the second group of inflow data, regions of highdensity occur downstream of the fuel inflow, with low contour values near the base region. Thesignificant difference between these data contours and those of the second data set is theshifting of the mild low density troughs that previously crossed the fuel and oxidizer streams.

A plot of the species D(O) and D(1) density contours, Fig. 28, show the product speciesexisting in a fairly narrow band of the mixing region. with the contours displaced slightlydownward toward the fuel nozzle centerline. The magnitudes of the D(1) contours are thelargest of all the contours obtained for the lasing calculations, with the D(O) contours beingsomewhat less than the peak values obtained for the second data set. and for the same dataset with no base relief. It is noteworthy, however, that the maximum density contours in Fig.28 occur at the outflow boundary, which suggests that a computational region extendingfurther downstream may be needed.

Sample problem 3In the previous two examples, adaptive solutions have been obtained for two different

configurations and a variety of inflow conditions. All of these results have been based on an

Contour Line Data: min= 0.30 E-5 max= 3.30 E·5 inlerval= 0.5 E·5 (g/cm3)

Fig. 26. Adaptive mesh and total mass density contours for a data set three. a viscous base reliefsolution without chemistry.

~CootourLine Dat.a: min= 0.10 E-6 max- \.10 E-6 interval= 0.2 E-6 (gfcm3)

(a)

~

~ootTlZ

::c::>enen..:lCo'-

J

(gfcm3)

---.:..-.....--~c:.~~

.,

Contour Line Data: min= 0.70 E-7 max= 5.70 E-7 interval= 1.0 E-7i

(gfcm3)

ooC)

<>

"-HC>

Contour Line Data: min= 0.30 E-5 max= 3.30 E-5 interval= 0.5 E-5

Fig. 27. Adaptive mcsh and total mass density contours for data set thn:c. aviscous lasing solution with base relicf. Fig. 28 .

(b)

Spccies contour plots for species (a) D(O) and (b) D( I) for data setthree, a viscous lasing solulion with base relief.

,.. .~

\ .

Chemically· reacting radiative flows

Contour Line Data: min= 2.10 E·5 max= 3.50 E·5 interval= 0.2 E·5 (g/cmJ)

Fig. 29. Adaptive mesh and total mass density contours for data set two with error indicatorcalculations based Oil the total energy. A viscous lasing solution without base relief.

987

error indicator calculated with the total density as the prime variable. Even though the resultsseem to be quite reasonable, other field variables are also available which may provide similarresults with less computational effort.

For the lasing calculations. the chief region of interest is the sub-area in the lasing cavitywhere the chemical reactions and radiation are taking place. These regions are characterized byhigh concentrations of vibrationally-excited species and possibly higher temperatures due to theexothermic chemical reactions. This observation suggests that a particular species density or thetotal energy may provide for a more efficient, localized refinement sequence in the lasingcavity. It should be noted. however. that estimates based on these other variables may notproduce results as good as the total density if shocks or regions of recirculation are present.

Using the second group of inflow data in Table 4 and the no-base-relief configuration of Fig.5, all adaptive solution was calculated using the total energy to evaluate the error indicators.

•• The adaptive mesh and mass density contours obtained for this solution are shown in Fig. 29.Surprisingly. the resulting mesh is almost identical to the result obtained using the total massdensity to evaluate the error (see Fig. 12). The density contours are also quite similar to theresults obtained previously with low density regions occurring downstream of the nozzleinterface and high densities occurring downstream of the fuel inflow.

As a second example using another error indicator. an adaptive solution was calculated forthe same set of conditions as above. with adaptivity calculations based on the species densityD(O). The resulting mesh and total density contours are shown in Fig. 30. As anticipated, therefinement has now been confined almost exclusively to the interface region between the twonozzles. with no refinement occurring near the oxidizer centerline. Comparing the densitycontours obtained here with the previous results. one finds a high degree of similaritydownstream of the nozzle interface and fuel inflow region. but much less detail is now availabledownstream of the oxidizer inflow.


Contour Line Data: min= 2.10 E·5 max= 3.50 E-5 interval= 0.2 E-5 (g/cm3)

Fig. 30. Adaptive mesh and total mass de:nsilY COIIIOur for data set lWO with error calculations basedon species D(O). A viscuus lasing solulion without base: rclid.

.'

'.,

Sample problem 4The final example presents an adaptive solution for a much more complex computational

region, with the domain beginning just downstream of oxidizer and fuel nozzle throats. Here. aslightly different no-base-relief geometry has been considered with the fuel nozzle exit planenow approximately the same width as the oxidizer exit region. An initial mesh consisting of 74finite elements was constructed with two elements spanning the oxidizer exit plane, twoelements spanning the fuel exit plane, and ten downstream elements in the laser cavity. Withthis initial very coarse mesh. a first level uniform refinement was imposed at the outset whichincreased the number of elements in the mesh before adaptation to 296. This initial refinementwas performed a priori to any lasing calculations to provide more detail of the flow conditionsentering the laser cavity. In conjunction with this new geometry, the error indicator waschosen for species D(O), which should concentrate the refinement more in the laser cavityregion, rather than the nozzle throat area.

The inflow data for this configuration was assumed to be uniform with the intlow conditionstaken to be those of data set two in Table 4. All initial downstream conditions were set equal tothe inflow conditions, with average flow properties taken at the interface points between thetwo inflow streams.

The adaptive mesh and density contours obtained for this configuration are shown in Fig. 31.As expected. the majority of refinement has been restricted to the lasing cavity with a minimalamount of refinement upstream of the nozzle exit plane. Computed species contours areillustrated in Fig. 32. and a normalized error indicator distribution is shown in Fig. 33.


Contour Line Data: min = 0.15 E·5 max =2.85 E·5 Inlerval = 0.3 E·5 (g/cm3)

f.)

Fig. 31. Adapuve mesh and total mass density contours for a computational region beginning justdownstream of the nozzle throats. A viscous lasing solution for data set two.

Contour Une Data: min = 0.50 E-? max = 8.50 E·? Interval = 1.0 E-? (g/cm3)

Contour Line Dala: min c 0.30 E-g max - 6.30 E·g

t.

Fig. 32. Species contour plots for species (a) D(O) and (b) D(I) for an expanded computationalregion .

...,

Fig. 33. (a) A dynamically refined mesh and (b) NormalizcQ species error indicator distribulion for anIHype adaptivc grid.

990 1. M. BASS and 1. T. ODEN

5. CONCLUDING COMMENTS

The preliminary results obtained in this study clcarly showed the great potential of adaptivemethods. In sample problcms, using of adaptivc methods led to substantial savings incomputational cffort. In sevcral cases, rcsults were obtained on an adapted (optimized) mcshusing, for example, 2000 elcments that were of equal quality as those obtain cd on a uniformfine mesh of 6250 elements. Moreover, the results also revealed quantitative information on thedistribution of error in the mesh whereas in conventional schemes one may only guess whenand where the solution is good or bad.

It should bc mcntioned that, with adaptive techniques, it may no longer make scnse togenerate finc meshes for computational fluid dynamics studies. What is marc appealing is todefine only an initial crude mesh with sufficient structure to define major gcometrical featuresof the flow domain. Then, through the use of adaptive schemes, the computer canautomatically produce a near optimal mcsh structure that will yield everywhere a preassign cdlevel of accuracy.

There are still many issues that should be investigated in the use of multigrid techniques andmoving mesh algorithms in conjunction with the adaptive strategy, the development offlux-limiting and flux-vector-splitting schemes to enhance solution quality near shocks andcontact discontinuities, the development of asynchronous time-marching schemes to optimizetime step selection, further studies on useful a posteriori error estimates for reacting flows andfurther refinemcnt of the data management schemes for adaptive refinement. The methodsdeveloped are already applicable to transient problems, and by applying time-dependentadaptive strategies that produce an evolving sequence of near optimal meshes, the simulationof the effects of start-up transients and time-dependent boundary conditions could easily beadded to this analysis capability. It must be emphasized that the adaptive schemes and theexplicit solution techniques are easily extended to three-dimensional problems.

The "optimal" fluid dynamics code for lasing applications should produce the best possibleresults for a given computational effort for a general class of lasing problems. We believc thatadaptive methods offer the only realistic hope for attaining such an optimum. Moreover, onceit is decided that it is desirable to attempt to achieve an optimal solution to a given problem,most of the fluid dynamics algorithms currently available are doomed to failure; they cannotcope with the special requirements of data structure and mesh topology needed for goodadaptivity.

In view of the special requirements of adaptive schemes and of the special requirements ofany flow simulation which realistically models complex now field geometries, thcre are clcarlyseveral basic criteria that must be met by modern algorithms if progress toward optimal codesis to bc made. Some of thcsc are listed as follows:

I. Mesh independence (unstructured meshes). The fluid dynamics algorithm should bcessentially independent of a global coordinate system and global mesh geometry. It shouldfunction on unstructured meshes and focus on simulating local properties of the flow as theyevolve.

2. Arbitrary geometries. The method should be free of any limitations due to complexgeometrical features of the !low domain or of the boundary conditions on the flow.

3. Accuracy. The method should be capable of delivering high accuracy. Certainly, if thescheme is to bc adaptive, one must also be able to expect a payoff in improved quality of thesolution once the structure of thc approximation has bcen altered.

4. Mathematical basis. Thc algorithms forming the basis of the method must rest on a solidmathematical basis, else questions of accuracy and a posteriori error estimation aremeaningless.

5. Robustness. Thc method must be numerically stable and somewhat inscnsitivc todistortions of the mesh and to singularities.

6. Supercomputing. The code structuring and basic algorithms must lend themselves tomodcrn supercomputing environments. Thc algorithms, for example. should be vectorizable orbe appropriate for implementation on array or parallel processors.

7. Computational efficiency. The algorithm, of course must be computationally efficient. Thisrcquiremcnt can be related to the supercomputing requirement, which is listed because it is

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perceived that a supercomputing environment may be essential for coping with the datamanagement problems of adaptive schemes for very large simulations.

Most existing finite differences schemes fail to satisfy criteria 1-4. In our opinion, only finiteelement methods possess the basic features that can hope to fulfill all of the above criteria. Thisdoes not preclude the use of difference approximations in time. or difference methods for thecomputation of error indicators, or boundary element methods for modeling viscous boundarylayers. or spectral-element methods in a subspace-enricbment adaptive scheme, or even finitevolume methods in cases of solutions with low regularity, but these criteria do suggest that it isdesirable to insert elements or distort clements at arbitrary regions in an irrcgular mesh so asto reduce local error. When one considers requirements 1-4, finite element methods seem tobe a natural starting point for the development of modern computational fluid dynamicsschemes which satisfy the criteria listed.

It must also be noted that there does not exist a sct of algorithms and data managementstrategies that satisfies all of these criteria in a fully satisfactory way. Nor do we expect there tobe a unique optimal code as defined by these criteria. However, we do feel that many of ourdevelopments discussed in this report make a significant first step toward an optimal code.

Acknowledgemelll-The support of this effort by Kirtland Air Force Base under Contract F-29/i0l-86-C-0246 isgratefully acknowledged.

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(Received 17 December 1987)

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