jjjjjjj - Defense Technical Information Center swerr-tr.v2 37 jjjjjjj solution of fluid dynamic...
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* ~~~SWERR-TR.V2 37 jjjjjjj SOLUTION OF FLUID DYNAMIC EQUATIONS FOR GUN TUBE FLOW BY THE METHOD OF WEIGHTED RESIDUALS , PART I. UNSTEADY COMPRESSIBLE BOUNDRY LAYERS TECHNICAL REPORT Dr. Rao V. S. Yalamanchili and Philip D. Benzkofer D D Cx June 1972 AUG 9 1972 RESEARCH DIRECTORATE B WEAPONS LABORATORY AT ROCK ISLAND RESEARCH, DEVELOPMENT AND ENGINEERING DIRECTORATE U.S. ARMY WEAPONS COMMAND NATIONAL TECHNICAL __INFORMATION SERVICE Approvee for public release, distribution unlimited.
Transcript of jjjjjjj - Defense Technical Information Center swerr-tr.v2 37 jjjjjjj solution of fluid dynamic...
* ~~~SWERR-TR.V2 37 jjjjjjjSOLUTION OF FLUID DYNAMIC EQUATIONS
FOR GUN TUBE FLOW BY THE METHOD OF WEIGHTED RESIDUALS
, PART I. UNSTEADY COMPRESSIBLE BOUNDRY LAYERS
TECHNICAL REPORT
Dr. Rao V. S. Yalamanchiliand
Philip D. Benzkofer D D Cx
June 1972 AUG 9 1972
RESEARCH DIRECTORATE B
WEAPONS LABORATORY AT ROCK ISLAND
RESEARCH, DEVELOPMENT AND ENGINEERING DIRECTORATE
U. S. ARMY WEAPONS COMMAND
NATIONAL TECHNICAL__INFORMATION SERVICE
Approvee for public release, distribution unlimited.
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I ORIGINATING ACTIVITY (Corporat* author) 0. REPORT SECURITY CLAS ICATIONU.S. Army Weapons Command UnclassifiedResearch, Dev. and Eng. Directorate 2b COUP
Rock Island, Illinois 61201.3 REPORT TITLE
SOLUTION OF FLUID DYNAMIC EQUATIONS FOR GUN TUBE FLOW BY THE METHOD OFWEIGHTED RESIDUALS (U)PART I. UNSTEADY COMPRESSIBLE BOUNDARY LAYERS
4 DESCRIPTIVE NOTES (Type of rep ot end Incluaivo d tee)
S AUTNORIS) (Pitat name, midce Inltil, leatname)
Dr. Rao V. S. Yalamanchili and Philip D. Benzkofer
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The overall gun tube heat transfer problem applicable to any weapon, ammunition, andfiring schedule was analyzed. Toward this goal, the solution to one of the fiveproblems involved was investigated by personnel of the Research Directorate, WeaponsLaboratory at Rock Island. The governing equations of unsteady compressible boundarylayers are a system of nonlinear parabolic partial differential equations with threeindependent variables. The transverse coordina, is modified to absorb the compres-sibility effect. The stream function is introduced to satisfy the continuity and alsoto eliminate one of the dependent variables. The methoa of weighted residuals is usedto reduce by one the number of independent variables. The functional in approximateso jtion form is chosen on the basis of the asymptotic solution of the steady differ-ential equations for large values of the spacelike coordinate. The error functions,consequently, occur in the solution form for boundary layers. The method of Galerkinis used as the error-distribution principle. All integrations across the boundarylayer are performed analytically. The Method of Lines is used to reduce the resultingpartial differential equations in two independent variables to an approximate set ofordinary differential equations. This procedure enables one to solve for derivativesby the reduction of a matrix with elements not wore than the number of undeterminedparameters introduced into the solution form. Finally, the solutions are obtained bya fourth order Runga-Kutta integration scheme, The typical output contains not onlyprofiles of velocity components, temperature, and density but also various boundarylayer parameters and convective heat transfer coefficient. The numerical results arein agre enl with Hall and Blasius results with even one term in approximate solution |
-form. (U)( Yalamanchili, R.V.S. and Benzkofer, P.D.)D OU 1o 17 OjART=,U6S. Uncl ARMY USE
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14 LINK A LINK 5 LINK CKEY WOROS - - - -KYOR ROLE WT ROLE WT ROLE WT
1. Gun Tube Heat Transfer
2. Unsteady Compressible BoundaryLayers
3. Nonlinear Partial Differential
Equations
4. Method of Weighted Residuals
5. Galerkin Method
6. Method of Lines
IIII
UnclassifiedSecurity C,..si.,,o°
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AD
RESEARCH DIRECTORATE
WEAPONS LABORATORY AT ROCK ISLAND
RESEARCH, DEVELOPMENT AND ENGINEERING DIRECTORATE
U. S. ARMY WEAPONS COMMAND
ITECHNICAL REPORT
SWERR-TR-72-37
SOLUTION OF FLUID DYNAMIC EQUATIONSFOR GUN TUBE FLOW BY THE METHOD OF WEIGHTED RESIDUALS
PART I. UNSTEADY COMPRESSIBLE BOUNDARY LAYERS
Dr, Rao V. S. Yalamanchili
andPhilip D. Benzkofer
June 1972
DA ITO611OIA91A AMS Code 501A.11.844
Approved for public release, distribution unlimited.
i(Q
/
ABSTRACT
The overall gun tube heat transfer problem applicaL'eto any weapon, ammunition, and firing schedule was analyzed.Toward this goal, the solution to one of the five problems
involved was investigated by personnel of the ResearchDirectorate, Weapons Laboratory at Rock Island. The govern-ing equations of unsteady compressible boundary layers area system of nonlinear parabolic partial differential equa-tions with three independent variables. The transversecoordinate is modified to absorb the compressibility effect.The stream function is introduced to satisfy the continuityand also to eliminate one of the dependent variables. Themethod of weighted residuals is used to reduce by one thenumber of independent variables. The functional in approxi-mate solution form is chosen on the basis of the asymptoticsolution of the steady differential equations for large valuesof the spacelike coordinate. The error functions, conse-quently, occur in the solution form for boundary layers. Themethod of Galerkin is used as the error-distribution principle.All integrations across the boundary layer are performedanalytically. The Method of Lines is used to reduce the re-sulting partial differential equations in two independentvariables to an approximate set of ordinary differentialequations. This procedure enables one to solve for deriva-tives by the reduction of a matrix with elements not morethan the number of undetermined parameters introduced intothe solution form. Finally, the solutions are obtained bya fourth order Runga-Kutta integration scheme. The typicaloutput contains not only profiles of velocity components,temperature, and density but also variuus boundary layerparameters and convective heat transfer coefficient. Thenumerical results are in agreement with Hall and Blasiusresults with even one term in approximate solution form.
ii
CONTENTS
Page
Title Page
Abstract ii
Table of Contents iii
Nomenclature v
I. Introduction 1
2. Statement of Problem 4
3. Theoretical Analysis 7
3.1 Method of Weighted Residuals 10
3.1.1 Approximate Solution 11
3.1.2 Weighting Functions 13
3.1.3 Integral Equations 15
3.2 Method of Lines 28
3.3 Boundary Layer Parameters 29
4. Numerical Analysis 33
5. Sample Problems 36
5.1 Rayleigh-Blasius Flow on a Flat Plate 36
5.2 Shock-Induced Boundary Layer 41
6. Results and Conclusions 44
Literature Cited 54
iii
CONTENTS
* Page
Appendices
A. Evaluation of Integrals 60
B. Listing of Computer Program 65
Distribution 95
DD Form 1473 (Document Control Data R&D) 98
iv
NOMENCLATURE
A1 , A2, B1, B2 Functions of x and t as defined inEquation 3.11
C = Coefficient in dynamic viscosity-temperature
relationship
Cf Skin friction coefficient
Cp Specific heat at constant pressure
D : Inner diameter of the tube
f's, g's = Functions in the ordinary differentia7equations
g = Gravitational constant
h Convective heat transfer coefficient
K Thermal conductivity
L Length of the plate
Nu : Nusselt number
p = Pressure
Pr - Prandtl number
Prt Turbulent Prandtl number
Q = Heat flux
r = Function of similarity variable (n)-Equation 5.12
R Gas constant
Re Reynolds number
s Function of similarity variable (n)Equation 5.12
v
/!
NOMENCLATURE
St Stanton number
t Time
T - Temperature
u = Tangential velocity component
u + Dimensionless tangential velocityEquation 2.7
v Normal velocity component
V = Velocity
x = Longitudinal coordinate
y Transverse coordinate
Y P dYIPO
0
y = Dimensionless transverse coordinate
Bly
=r ( 2 e-m'dmerf(bly) /:- (0
= : Stream function
= Approximation to p in terms of ,' 2, 3
6 = Difference in temperatures between gas and wall
p = Density of gas
vi
NOMENCLATURE
y - Specific heat ratio
= : Shear stress
V Kinematic viscosity
= Eddy viscosity
- Dynamic viscosity
= Covolume of propellant gas or similarity
= - Exponent in dynamic viscosity-temperaturerelationship
61,62, 2 = Coefficients defined in Equation 3.11
= Boundary layer thickness
6d : Displacement thickness
6ed Energy-dissipation thickness
6n Enthalpy thickness
6 = Momentum thicknessm
u = Velocity thickness
X = Dimensionless x-coordinate
Subscripts:
0 : Reference quantity
1 = Outer edge of the boundary layer
b = Behind the shock
w = Wall conditions
vii
NOMENCLATURE
Superscripts:
= Differentiation with respect to time
= Differentiation with respect to similarityvar 4able
viii
1. INTRODUCTION
As the projectile in a weapon moves ahead because of thehigh-pressure gases created by the burning propellant, thepropellant gas will be set into ,notion starting from rest.Since the governing equations of fluid dynamics for manyproblems of interest are a system of nonlinear partial dif-ferential equations which are dominated by real gas and non-equiiibrium effects, no general solutions exist by whicharbitrary initial and boundary conditions are allowed.Therefore, examination of the flow field and the subdivisionof the overall problem by consideration of the dominantfeatures only seem appropriate The ultimate objective isto establish a capability to perform overall heat-transferanalysis for any given dimensions of the weapon and forspecified propellant characteristics, Toward this goal,the propellant gas convective heat-transfer prnblem! isdivided into five subproblems. (1) generation of thermo-chemical properties for any given propellant, (2) transientinviscid compressible flow through the gun barrel (coreflow), (3) unsteady viscous compressible flow on the boresurface (boundary layers), (4) transient heat diffusionthrough the multilayer gun tube, (5) unsteady free convectionand radiation outside the gun tube.
As the propellant burns, more and more hot gases will begenerated, Typical gas in a chamber contains a temperatureof 5000'R and a pressure of 50,000 psi. The thermochemistryof propeilants involves determination of chemical compositionof gases either by finite-rate chemistry or by chemical-equilibrium chemistry and the derivation of gas propertiesfrom the composition Thermochemical properties of typicalpropellant gases are being predicted by use of a NASA-LEWISthermochemical program, The gases were highly toxic.Typical composition of the gases: M18 (CO = 0.41, Hl = 0.19,H20 = 0.16, N. 0.1, CO 0.08) and IMR (CO = 0,43,H2 = 0.12, H O 0 21, N 0 12, CO, z 0-12) IMR is betterthan M18 as far as combustion is concerned, but IMR is still along way from possible complete combustion The consequencesof incomplete combustion are muzzle flash, smoke, ,nd fire inaddition to low elficiency W'th th's program, one can com-pute not only chemical composition of gases but also specificheats, gas constant, and so forth as functions of pressureand temperature.
In instances of the unsteady core-Ilow problem, uniformdensity, uniform temperature, and lmner velocity gradientare commonly used in a gas flow betwee'n the breech and thebullet. Since the governing equations of this problem are ofthe hyperbolic type, they can be solved by the well-known
4
method of characteristics. Some of the results obtained bythis method were discussed ' before. None of these assump-tions is reasonable until after peak heating occurs. Ingeneral, however, the linear velocity gradient assumption isbetter than the uniform density assumption.
The boundary layer problem can also be interpreted asforced convective heat transfer in guns. The magnitude ofconvective heat transfer in guns is large primarily becauseof the high values of gas densities that exist and becauseof the large gas-to-wall temperature differences in additionto the larger gas flow velocities which constitute the convec-tive heat transfer driving potential. Experimental data arecommonly correlated with the various analytical or empiricaltechniques. The most popular method for heat transfer corre-lation purposes derives from the postulates of Nordheim, Soodakand Nordheim.4 These authors theorize that the propellant gaswall shear friction factor is dependent upon only the gun sur-face roughness considerations. The justification for elimina-tion of the Reynolds' number as a parameter is based on orderof magnitude estimates of boundary layer thickness by use oflaminar boundary layer considerations. The recommended formof the friction factor is thus dependent upon only the ratioof surface roughness to barrel radius. Consequently, thefriction factor is assumed to be independent of space or timewithin a given gun barrel. Reynolds' analogy is subsequentlyused, and the heat transfer coefficient becomes simply pro-portional to gas density, velocity, and specific heat. Totalcalorimeter data have subsequently been rationalized in termsof the value of the friction factor that, with internal bal-listic considerations, obtains the spatial varidtion of heatload to the gun barrel derived from a single shot. Examplevalues of experimental friction factor (Cf/2 = Tw/PV 2 ) that
were reported vary from 0011 to .0035. Geidt s assumes avalue of 002 and finds that his measured heat flux is pre-dicted generally within about a factor of 2. Other writershave interpreted pressure gradients within the gun barrelin an attempt to evaluate wall shear for the application ofReynolds' analogy for heat transfer.
Another approach for correlation of experimental datais based on the Dittus-Boetter6 relation for steady, fullydeveloped, turbulent pipe flow. This relation is also judgedto be inaccurate in several references. Cornell AeronauticalLaboratory' proposed a combined analytical-experimentalapproach based on steady, fully developed, pipe flow concepts.
The state of the art in unsteady boundary layers andturbulent models is quite limited. A symposium' was held on
2
t
unsteady boundary layers at Laval University (1971) under theauspices of international Union of Theoretical and AppliedMechanics. However, their proceedings are still in press.Patel and Nash9 discussed solutions of unsteady two-dimensionalincompressible turbulent boundary layer equations by explicitfinite-difference techniques Akamatsu il pointed out somekinds of unsteady boundary layers induced by shock waves in ashock tube. Woods!! identified some industrial problems (suchas pulse turbine, reciprocating engine, emergency blowdown ofa chemical plant autoclave system, high-speed train enteringa tunnel, and exhaust system of an internal combustion engine)associated wgth unsteady boundary layers, Foster1 2 solvedunsteady isothermal turbulent boundary layers with severalapproximations by the integral method and the method of char-acteristics, Anderson and Dahm! 3 solved unsteady laminarboundary layer equations by the integral matrix procedure.Shelton "a developed a solution procedure by combination ofintegral methods and finite-difference methods for turbulentboundary layers that are in compliance with Croco-Lees51 rela-tionship for temperature. At the end, the Chilton-Colburn5 2
analogy was used to compute convective heat transfer coeffi-cients.
Dahm and Anderson" formulated an analytical boundarylayer procedure based on the compressible time-dependentboundary layer momentum integral equation by the simplerCroco-Lees relationship and the method of characteristics.Convective heat transfer was evaluated based on the Chilton-Colburn analogy of the energy boundary layer to the momentumboundary layer. Since the momentum and the energy boundarylayer equations are dissimilar in an accelerating flow, anapproximate solution Df the energy integral equation isexpected to yield better heat transfer results than applica-tions of the Chilton-Colburn analogy to an approximate solu-tion of the momentum integral equation
The solution of transient heat diffusion through gunbarrel walls was established quite satisfactorily by ana-lytical,-t finite-difference- and finite-element ,s19,20techniques. These are quite good for the purpose of estab-lishing propellant gas convective heat transfer coefficients.
The unsteady free convection and radiation around guntubes was discussed.' The radiation and convection contri-butions were of the same order of magnitude. The estimationsbased on pure convection show that the flow around the guntube is still in the laminar range. Since the surface tem-peratures change quite rapidly for automatic weapons, thegoverning time-dependent nonlinear partial differential equa-tions with three independent variables were solved by an ex-plicit finite-difference scheme The stability criteria were
3
established by Von Neumann and Dusinberre22 methods. The con-vective heat transfer coefficients can vary as much as 100 percent from the minimum value because boundary layers are thin-ner on the lower half of the gun tube and thicker on the upperhalf of the cylindrical gun tube.
2. STATEMENT OF PROBLEM
The present investigation concerns primarily the formu-lation and the solution of transient viscous compressible flowon the bore surface. However, this is affected extensively byunsteady core flow and unsteady heat diffusion through the guntube due to boundary conditions. The flow characteristics areunknown for gun barrel flows. The experimental data are lack-ing because of the moving bullet. This obstacle may be over-come provided one takes advantage of the similarities betweenthe moving bullet (small mass) and a moving shock in a shocktube. Therefore, shock tube data should be collected andanalyzed for possible use in predicting gun barrel flow char-acteristics.
As the propellant gases expand behind the projectile, aboundary layer forms at the breech end and thickens as theflow proceeds downstream. An unusual feature of the velocityboundary layer is that it disappears as the bullet is approachedsince all fluid at the base of the bullet must be moving atbullet velocity. Mathematically, this amounts to the require-ment of an additional boundary condition at a downstream loca-tion. The numerical techniques applied to most boundary layerproblems require the specification of profiles at the upstreamend of the flow and allow a "marching" along the flow direction.For the usual time-dependent boundary layer problem, an initialcondition o describe the boundary layer flow at time zero andboundary conditions as functions of time are necessary. Nodownstream condition is added. The question then logicallyarises as to how the boundary conditions, at both ends of theflow, can be satisfied in any one analysis. If the analysisis carried out from both ends of the flow, the results may notmatch anywhere in the middle of tLe flow. The compatible con-ditions are required before one can accept the results in themiddle of the flow.
The laminar boundary layer becomes unstable if the Reynolds'number becomes sufficiently high, thus small disturbances willbe amplified causing transition to a turbulent type of boundarylayer. For a flat plate with zero pressure gradient, the laminarboundary layer has been experimentally shown to be quite stablefor the length Reynolds' numbers Rex upward to about 80,000, and
this laminar boundary layer can extend to a Reynolds' number of
4
several million if the free-stream turbulence is very lowand if the surface is very smooth, For engineering calcula-tions unless other information is available, transition willbe assumed generally to occur in the 200,000 to 500,000 range.These figures are only approximate and may be gooJ for asmooth surface with a fair amount of free stream turbulence.
The length Reynolds' number Rex is not very convenient
for a transition criterion since it is based on a constantfree stream velocity and may not be meaningful if it is al-lowed to vary with axial coordinate (x) such as the acceler-ating flow in a gun barrel In such circumstances, it ispreferable to have a local parameter such as momentum thick-ness (6 m ) instead of x. If a critical Reynolds' number Rex
of about 360,000 is chosen as transition criterion for a con-stant free stream velocity, the equivalent criterion foraccelerating flow becomes Re6 = 0 664 ITWxJ = 398.4
The Reynolds' number, based on local distance, does nothave any boundary layer characteristics, whereas the Reynolds'number based on momentum thickness does represent the impor-tant parameter of the boundary layer, Since transition to aturbulent boundary layer is dependent strongly upon the growthof the boundary layer, the Reynolds' number based on momentumthickness is logically considered te establish transitioncriteria.
The rate of heat transfer from the hot propellant gasesto the barrel is controlled by jevelopment of the boundarylayers. The flow in the gun barrel boundary layers could belaminar, transitional or turbulent in nature. The type ofboundary layer at a particular cross section at any instantneed not be the same as at another instant. Since the flowmust start from rest and :iust also satisfy zero boundarylayer thickness at the bjllet base because of the scrapingaction of the bullet, laminar flow always exists in someparts of the gun barrPI boundary layers
Flow in a laminar boundary layer will eventually becomeunstable as the Reynolds' number is increased, The boundarylayer thickness, slin friction, and heat transfer increasemore rapidly in turbulent flow than in laminar flow. Theeddy viscosity is ,-he dominating mechanism for such increases.The boundary layer flow can be turbulent somewhere in themiddle of the flow between the breech and the bullet base. Atransitional region should exist between the laminar and theturbulent regions, However, because of limited knowledgeabout transitional regions, the flow will be assumed to changesuddenly from laminar to turbulent flow at a time and place
determined by the well-known laminar-turbulent transitioncriteria discussed above. Therefore, an analysis of theunsteady boundary layer is needed for laminar and turbulentboundary layers.
The analysis of unsteady compressible boundary layers onthe bore surface is one of the most difficult problems due tothe limited state of the art and also to the existence oflaminar, transitional and turbulent regimes within the bound-ary layer. The governing equations of unsteady compressibleboundary layers are a system of nonlinear partia' differentialequations of parabolic type with three independent variables.These are given below:
Continuity:
+ I (pu) + (pv) = 0 (2.1)
Momentum:
au axu au a a auIT.p v@ x [(+, (2.2)
Energy:
p C (3T + U DT+ T) +=2p (V1) 2)p at ax + v 3 X + x
+L [(K + pPrS--) T] (2.3)
Equation of State:
p l1 - n) = RT (2.4)
Boundary Conditions:
y = o : u = o, v = o, T = T w(x, t)
y=c : u = ui(x, t), T = TI(x, t) (2.5)
x = x0 : u = u u (y, t), T = Tu(Y,t)
6
Dynamic Viscosity:
: CT (2.6)
The following eddy viscosity (e) model may be used for turbu-lent boundary layers:
+y V g -T w/py
u" (2.7)UW -
/g /IP
s/v = -,Ir,du /dy
E/v = - -- for o < y < 5ro
C/V = 5 r l for 5 < y < 30
Y+(l-ro)land e/v 2.5 for 30 < y <
The objective is to obtain the dependent variables u, vand T as a function of x, y and t subjected to the boundaryconditions given by Equation 2.5. This is discussed in sec-tions 3 and 4.
3. THEORETICAL ANALYSIS
Various methods exist for the solution of steady boundarylayer equations. Thp classical Von Karman-Pohlhausen integralmethod is popular because of the quickness and simplicitycharacteristics. For particular free-stream distributions suchas flow similarity, the partial differential equations can bereduced to ordinary differential equations by similarityvariables. At the end, the results are obtained by numericalintegration. Instead of solving partly analytically andpartly numerically, one can also solve partial differentialequations of boundary layer by finite-difference schemes. Asan alternative and also as a reduction in computational times,one can use the method of weighted residuals for the solution
7
//
of boundary layer equations. The last approach is used in
the present investigation.
The dependent variables are u, v, and T. The indepen-dent variables are x, y, and t. One can easily reduce one ofthe dependent variables such as v by introduction of a streamfunction, p. The equation of continuity is satisfied auto-matically if the velocity components are denoted by the fol-lowing equations:
p ay
vp9o ( + a )(3.1)P ax a t(.1
y R dy0
where p0 represents a reference density for nondimensional
purposes.
The Howarth and Dorodnitsyn transformations among othersare an important class of transformations specifically designedfor compressible fluids. The purpose of these transformationsis to remove the density from some of the boundary layer equa-tions and to present the equations in a form closely resemblingthe incompressible boundary layer equations. After these inves-tigations have been pursued, the transverse coordinate is modi-fied to absorb the compressibility effect. The following equa-tions can be obtained by the specializing of Equations 2.2 arid2.3 to the boundary layer edge conditions (y
au +u
I ax ax3TI @T P
c 7-+ u I) - +u (3.2)1 p at 1 T - at ax
The momentum eqw,-tion is reduced to the following formif Equations 3.1 and 3.2 are utilized in addition to the identity
ay Poay
8
+ u + [(v+e) a(- . (33)p at 1 lax + TY a
The pressure gradient terms in the energy equation canbe expressed in terms of free-stream quantities:
DT DT
+ u p C ( +at ax p at u )-(u-u) p
au au
at I + u I )ax (3.4)
This equation is obtained by use of Equation 3.2.
A new dependent variable can now be conveniently intro-duced for temperature difference as
o = (T - T ) (3.5)
Substitution of Equations 3.1, 3.4, and 3.5 into theenergy Equation 2.3 yields the following equation for thevariable 0:
aeaa 0 +x a- T aTwaCx x +a @3 ' + Cp'at ay ax
a 2( K + PC 6) ae] (-V+E) (*.)2P
- a p0 Prt )j p 0 a32
p @T 3T p au au+, P -p3 - + u: a (u - ul) - 1 at I + u ax (3.6)
Further analysis of Equations 3.3 and 3.6 will be conductedin the remainder of this section.
9
3.1 Method of Weighted Residuals
The method of weightcd residuals unifies many approxi-mate method, ofthe solution of differential equations thatare In use day. For unsteady heat conduction, the finite-element method and the usual finite-difference method wereshown 2 3,z,18 9 to be special instances of the method ofweighted residuals with a general weighting function. InReferences 24 and 25 in a more formal way, variational prin-ciples proposed by several authors are all applications ofthe method of weighted residuals.
An excellent review (187 references) on the method ofweighted residuals was presented by Finlayson and Scriven.2 5
In literature, this technique is commonly called the errordistribution-principle. Usually, the method of weightedresiduals is used to reduce by one the number of independentvariables in any system of partial differential equations.However, some exceptions exist. For example, Kaplan andBewick 26 and Kaplan and Marlowe2 7 used the method of weightedresiduals to reduce the number of independent variables fromfour to two. When this procedure is combined with othernumerical methods, significant reductions in computer timeto obtain a solution is apparent.
The basic steps that are involved in any applicationof the method of weighted residuals are given below. Equation3.3 can be abbreviated as
S(xIyt) = 0 (3.7)
Let the following equation be an approximate solutionchosen to represent p. The selection ce approximatingfunctions 2 ill be discussed later. However, theseare usually chosen towsatisfy any known boundary or initialconditions with respect to the independent variable to beremoved.
= (B1(x,t),y) + 2 (B2(x,t),1Y) + 3(B3(x,t),7 ) (3.8)
Substitution of the approximate solution for p inEquation 3.7 yields the following relationship for the residualerror:
,(Bl(x,t),y) + p 2(B2 (x,t), ) + V13 (B,(x,t),y) -
R(B1 ,B2 ,B3 , ) (3.9)
10
The functions B B2, and B are to be determined suchthat, in some sense, the residual error approaches zero.This objective can be accomplished by multiplication of theapproximate Equation 3.9 by a set of three linearly inde-pendent weighting functions (W1, W2, W3 ) that are dependentupon B's and y. The weighted residual error is then inte-grated over y between o and ', and the result is set equalto zero.
i.e.,
TR(BI,B2,B 3,7) W,(B,(xt)y) dy = 0
0
R(Bl,B2,B ,y) W?(B (x,t),Y) dy = 0
0
IR(BB 2,B 3,Y) W3 (B 3 (Xt)) dy = 0 (3.10)
0
The resulting equations contain the unknown functions B,, B22and B3 with independent variables only x and t. Thus, theobjective of method of weighted residuals is achieved by re-moval of the dependency of one of the independent variables(y). In return, a set of equations (Equation 3.10) isderived for the determination of B, B2, and B The pprox-imate solution (Equation 3.8) need not be limiged to threeterms. In fact, increasing the number of terms in theapproximate solution increases the accuracy. However, threeterms may be enough for engineering accuracy.
3.1.1 Approximate Solution
The choice of approximating functions in an assumedsolution form is crucial in applying the method of weighted
residuals. No way presently seems to be available to selectthe approximating functions systematically for all problems.Selection of approximating functions remains somewhat de-pendent upon the user's intuition and experience, and thisis often regarded as a major disadvantage of method ofweighted residuals. Crandall 26 stated that the variationbetween results obtained by application of different weightingfunctions to the same approximate solution is much lesssignificant than the variations that can result from thechoice of different approximate solutions. Sometimes, onecan obtain the exact solution by use of method of weightedresiduals if the right choice is made in the selection ofthe approximate solution form.
The selection of approximating functions is still adefinite problem even though the method of weighted residualshas existed for more than 50 years. Several sets of approx-imating functions that satisfy boundary conditions are per-missible, and to choose one as the best is impossible. Theusual approach of selecting the approximating functions isbased on satisfying the governing differential equation onthe boundary in addition to satisfying the boundary con-ditions. However, Lowe29 established the form of the approx-imating function in the following manner:
If integration in the unbounded domain is to be accom-plished, then the integral must exist for large values ofthe argument. Therefore, the asymptotic expansion or vari-ation of the dependent variables should be established forlarge values of the space-like variables. This primarilymeans that the exponential order of the dependent variablesfor large values of the space-like variables should be de-termined. The approximating functions should be chosen toexhibit that some exponential order and, if possible, thecomplete asymptotic order. At the end, the approximatingfunctions should be made to satisfy the boundary conditionsand, in addition, these approximating functions shouldsatisfy the governing equation at the boundary. In effect,the residual will start and end at zero. The residual in
the interior of the domain will then be adjusted by someerror distribution-principle.
This is the approach used in the present investigation.
Following this approach, one can see that the error functionappears to be a fundamental form for approximating functionsfor forced flow boundary-layer problems. Therefore, thefollowing approximate solution forms were respectively chosenfor the dependent variables u (or f) and e,
12
u Q: erf (B,y) + 92 erf (B2y) +
0 : 6 erf (A 1 ) + 62 erf (A]Y) + (3.11)
Where A's and B's are unknown functions of the other two in-dependent variables x and t. The following must be true tosatisfy the boundary conditions fur large space-like variable.
1 + 2 + " " 1
61 + 6 2 + "'" : 1 (3.12)
3.1.2 Weighting Functions
With the weighting functions, various criteria thatare available are determined for the distribution of errorin the method of weighted residuals. Basically, five criteriaare available. The origin of these techniques and corres-ponding weighting functions are given in the following table:
Method Origin Weighting Function
Collocation Frazer, Jones and Divac - L'-ItaSkan 3"
Galerkin Galerkin3' (Equation (3.8)3B i = 1,2,3
Method of Moments Kravchuk 32 any complete set offunctions
(1 ,y,y2 . . .)
Method of Least Picone 33 (RSquares~-~-T. (Equation (3.9)Squares 1B
Subdomain Biezeno and Koch 3" Unity in that subdomainand zero elsewhere
13
To name a particular criterion as a best one is impossible.The choice may depend upon the problem to be solved, theassumed approximate solution form, the number of terms init, and also the parameter of interest in that problem.The following example concerns the Blasius problem of forcedflow over a flat plate [f'" + ff'/2 = 0, f(O) = 0, theassume solution form f' = erf (Bn)].
Dimensionless Dimensionless Displace-Wall Shear ment Thickness
Method f' (0) % error 6, % error
Exact .33206 0 1.7208 0
Galerkin .3532 6.29 1.803 4.66
Method of Moments .3631 9.33 1.753 1.87
Subdomain .3631 9.33 1.753 1.87
Collocation .3927 18.25 1.621 5.81
The results by use of the method of least squares is unavail-able because of the possibility of imaginary results forsimilar classes of problems. Moreover, quite complex weightingfunctions will result by this technique. The parameter ofinterest for heat-transfer purposes is that of wall shear.The Galerkin method resulted in least error for wall shear.Finlayson and Scriven" solved a convective transport problemand concluded that the Galerkin method predicted exactly thesame expression for Nusselt number other than a proportionalityconstant. In the instance of the Galerkin method, the numberof terms in an assumed approximate solution did not yieldsignificant differences in heat-transfer results. Thesecharacteristics provide a more desirable method for coupled,nonlinear, and complex partial differential equations. Fromthe examples cited, the selection of criteria is evidentlyunimportant. However, one should carefully consider theselection of functions in the assumed approximate solutionform.
On the basis of the examples cited above and the state-ments by Ames, 3" Finlayson and Scriven,2' Schetz, 36 Snyder,Spriggs, and Stewart, V and Kaplan, 30 the Galerkin methodis chosen for the present investigation.
14
No evidence is available to prove that the method isa superior technique for general problems. However, theGalerkin method was related, before, to variational cal-culus, and several proofs of convergence have been made forspecific applications.
The weighting functions (Galerkin) for assumed solutionforms (Equation 3.11) can be written as
2220e -B 2 Y
2 2
2e ' 2 (3.13)
The first two weighting functions are meant for themomentum equation, whereas the last two pertain to the energyequation.
3.1.3 Integral Equations
The analysis mentioned under the method of weightedresiduals can now be performed. The approximate solutionforms (Equation 3.11) and the weighting functions (Equation3.13) are already obtained. The objective of this sectionis to explain the transformation of the dependent variablesfrom p and 0 to B1 , B2 , A,, and A2 . The objective also in-volves the removal of the transverse independent variable Yfrom the transformed governing Equations 3.3 and 3.6. Thestream function i and some of its derivatives must be evalu-ated before one can apply the method of weighted residualsto the momentum Equation 3.3.
15
'f r-
IbI
c Nl '
I IbC-
-'1 -. *-
Noi
4I0 cC;i 4
c I 4P -"0
It cdt ca 7-0-i C
+ ILI
0C t C4+e.i ~ CL..ilC
ci~a~C N. -. x
Ia 0: +-.- I 0
+ ~ %i - CiNte N
N N%
CC
C4 to ~ - ;
I#9 - (a
co +
+ coc-
co co- ~ C
ZC. co
(4 jj ca
LIZ-
Ct + + LJ + (
rI: ~* ~IT_____ I 0
~ G~Iq-K ~ 4 ~T. _ 1jCi
AA. n= (Bd e+eA- e -, " -I -"
where C1 and C2 are determined by applying known conditions on y to i(y=O,p=O
C ) aA _ , e., A u. , no
- + + 2.a~A. AL2 A,1 _
+ +AL,
L,4 _1+_. ) _,,'a& , ', e A 9)
ti
2C -x
-x 8~ ~ L~ +J..c-x
aKa
11) 9 . JJ- ca B-1B " x . : .
+ 2,_ 2.,.-- .(, -C) + .+ -- ;e a,Se - % [2 -, -,2-. L
52, cXB -x -AW
- , + e _Ak +u. 8, eB_ +.. Ou,
B, / 8;' -°x,+
c) 2 1 n ., .e 1.+ 2 -a . A ,,L _ .8.,1 e -B?
: 16. 1
?B
,+ AA.,. L ' )+ e-;" Cj
plying known conditions on y to p(y=O, p=O) Knowing , one can obtain
Sc U, c B 3.1'"
-a z6
A)LGtt')~f-ajL-g -
,, e- 9, - + A x AL.I
113 161
-4..
AB~ Ic8 + -a I I i v ( ,,).BI( -Z ).,A
Ba 8-x u LB -x
b [3 1~
cn N
10~'
Cf di o
ri coiI(~s (
0 -g
'c0 - 0
c o~t5 +
Ih. do'
.~'
'0_ -e
14
4
~~~:~~ I I c -N
C'P' 0!
+ ' A ~
+ N ') if 1'4 Ile-, iL
+ 1+
cc.0. d. N
- -I.
II
4bI~+ -5+
+ I + ; ca+ io 00 - co
-(D1,4
4
-
cc 6~-I ~ i 44 0
II-
Pt °, ALI C , AL, R + ? P+RTo, OA, Re + .P +- RTw_./ at '0 6% R9 1+1.P+ RTw at R1,-+ rL.P + RTw C)
82
Combining all terms of the Momentum Equation and forming the residual, one obtains
A. 2 cs, - , i 2 _ e-B, + _ , ai,
A + T= -, + a B;-4(+ B - I 2. u.A2a_ -) ,B, -)e
T o. C tB, - i, c.( j,,
2 8,.a -(B a±-3 1nL! 274-- e + -e- .o a , /a2. u, e- + D,.
, , A ,c , rAL, B, -B, i 2 -a,%ji
, -8- e O.,.e
__ 2. 1
._ _a A. 8 1 + a J3 a , --a, -e a2 +- f- ee,, 7 ,% I.- ) B, e C e
s, cl.. k, -(,+ ~ a z BIBB(8Z 2 .ciBz +(_a nz) 2 kL8A- )A#e+-LI_ -
~ti lJJCagyx c ir 6A e~
Re,+LP+RTw cit RGO-i rtP +RTw at R, + rP+RTw,, 'c; a- rGP +R"
17.|
R! e + q.P + RT , .,,. (,_ ,[ .0Re~tP+R - du [3.20]
R9 1+ LP + RTw 8
n and forming the residual, one obtains
-ve B, v(Bi4a 't) at + -, W-, t-m - a,) -' g + a At ,-
1r~~r- a - . ,
-l 23. 2 8B
4222 CL .A.B 9
. B1~.. ..u.. # -- . (B z,,4 , e -( , a-"' Ba- .n.., ,u. -- -a ~J. , 8,A
-L . ..,, .,- } "-- e-- . _-, ,u , , ,e--
eB, - - - -
, - AL, 5 s e
.2 4e B , + - eq 2. 2 3 _ ,B -2 Bc .,, +X.,e-
S- -a e - e
R , , .,(A ,) +a , + + 8 RTw -B., Rs 2 2 -, ('Aa ) -a 2,
7-~ ~~ a7- TRG,+~~~~~ 8PRT e' 2AIQ RT 1 ~ G1 +q - T )
Bz Jt7.B
I) AkI
e a aA.18
C; F---
-D I c c c
4J4
dm -9
Ci.Ci . 0 .
S- ciCLWL
>0
0) 0. co.Old 00
+ 11' 4
10 'o rd "lerr- -T -- 4J 4> ci j1 I< If
Go + 'C
z0 -, ' 0 -
>10) 1 1
c~j-o +
: a - 0.~
co CC. (Ao
'- -0;+ +.- >)4M 4
4, go 10- -- ,
R 1 +rLP + RTw )- 41T )
Multiply Equation 3.22 by the weighting function A ,4 E and integrate over.:analytically. Those integrals not readily obtainable are solved analytically in AppelEquation 3.22 above, one obtains
.3 AL, 1 82 a-~ 8aI_ _
AS, -a -11, A4,-5.07 t: 2 ~ B~lt
at 2. J:i 2 + a a Ir +B2 S..2.n 2 :88,-B~ 4- B))(2B? Bt
+k tat (0.707)2 +I fI BQ~ AA 1. 77 77 ___
dr-r c x B,28. (8 + J2+BIT ir JT FLBaB)
-aa a aLT )a± B> -I.
-T, I) 9, + P~+JJ.A L? 7 =1T, BaB)( 2 2.B dr
71= -jx 77'77'i2 T= 77 L2~+) (Bi +f V/W2 a
BR, ta , '(p707 2) + +- +8)12+) +a JOAL B
2. __ 1 1 2da B, 4-aC
4- C2 B 4). A.O AJ, 2 . 23, _ _
AA. , 1 - B . + d 8 + 2 , -t
- 0.a 7 (aB~ BY)2 Bz + 13~ Ba (B 1 ) 28,2 ±B')' + r 7Eir *1
_2 4-__ 3- C) A n2 I . 2+ - -B 1 a 2 , AA., I B+ 2B -m7 .f-1 2 ,J. . j-(8,'+8 13 +8*a~ 2 irirJ- 28+BY rTrc)- , 2B
-a A J a.,a 2 -a a 1 __ 821
c)~d' xB~ (zl n.1,Re + 32) f.,2c-- Bt+ l
18 A
------ 2---------X + 2 ---.- 1 I
IB 8e - '+ P* -a 8. Ie8 ''R [3. 2 2]
:tion 7F-,AL, " and integrate over y. The integrals are evaluatedzbtainable are solved analytically in Appendix A. Solving for , in
] B2 a 8z + 1 , -a8AL, ) fa )
it aB,• 3'7B +: a tI) F ---2 O-z + Ba -,; i
+ aA, ta (-a, + -I B B
iB1 T *B / +(-+B ((2+,,~ '+ BI), ta L77;)*t (I) a) .
8118'? + ,- .;,. B, J T + -
k' 6~iIr' &x L (2(p2B +8J[) (a;r(a4-B[ 3' 2,+a _ [C I a I I B
____-- -R X1 M4' -M )+(a 2)82a/ B 2B)(2B . " irfl B2 3~ (2~-~ jj'e x (a+~
2 8 I12 I + 8 2
71F a BB ___ +_+_B
7) 6 (e+ ir~f ' -X~ [i (2E3 ~ V gr i3 (8>
. .. I I (87u., I I
cBi +8-f Bt flc)+T 81L~-
81+-.rJI- aa- (..... ... ..- sit 77 F5.
a3A_2 c ,,A ''c 4,3c
+ A AJ
1r 83 C) -
+C
<-.-aI+
cv -z- Lv
co +o L,.. +d
+ + c
co cv hCI .
< I -" o IV co+- +- + , .1
co + < .jELi.oC E- ID "
CO , CN + C.
.v - r-- .0'C'
0 --- co co co Cd-. L
,Nt F- 4 r.Cd . r I , p
+ 4- - +- t
cv + tv . . .- I
4- -l 0
co + co+
N+ N +N +co + co + + . : -t
-, ". E a_ Is
Cd. 4(1V ++ 10 0~
-LLb.
-C4 ~ - N c-. d cm
ON~I -- '0 jC.1
(M d I, + + I 04
I
+ .._ . Re I ALI , +i I_-ag aL8 ., rP+RT, I + ITir at Re,+rtP+RTw BjIAa+Br " +j ""at R,+-P4RT, 3,1 1rB
+_ . . I A ,I .. b, ri p "R G sT Az
AL (A ',' R+,+2P+ R T , B ,A,2 + .o., i + 2 B
-a f O' Z (At +2Ba)A .+B +Tr ' (A,"+28,) "7aO'T' (AzaA+28)J-
-r Aj(6 ,8 + ,2+8 4 ,,3 64B, (2 +A -)s a 3 IOLJI 2 B2)+(2B?++ A) 1 I05(2 B,) 3 + 210(2 B1)2 A' +
A1 [ 36.4oAB.-A) _
:~ ~ L 6 .r2( 2 Bt4#.I + 10 TL 2. 6.1.2. B l) , + A4,/'(B {I+PB) + 1
+A , 2, 2 ) + 3(2B-2)A+ + 728(Z 8,)A+ 38
A1" 2. 7[" (2 ,B,) 2520(2,a) 3 A' -f- 3024(BI)' A. + ;( 172) .(2-32(2_ __ _ __ _ __ _ _ B2_________AT-__9_2 a ,' AIO( l6
SrAS(a )-+ 20( 8 +)A. , + 8At) ,A A,_1__S__2___3_+_p-T L s , a 2PB -a B2"'? + -A L5'/ "1 8(+' 4- A?)+' 2 /
A, A l. [9 45 ~a 2 ) ' + ZS2 0('aB it A 22 + 3 024(2B 8) 2A ' + 0 2~8(2 12) A ZG
+ 2Fa(A4B)]+ 2 atTW, -375'~) tahJ(~~ + 2~(B(~ .~]+!~
-l (l. + 2, e.' 2 2.) 2 2 .. .... ." ) 16- , - - '( - 8
2 I-T (-aa.__Fi_)Aa___2_
2. r i (2 __ __ _ __ _ + I r ,)(A. 28- i
+ +-LGi ,ITw I + 0 , A2Tw(AB+2A.r A.51' B2)B 4W A, 8.8)~
A~[Ai~+5B" +8 02oBl +82) A. +8 A 5 )] A[A Is(4B
19.1
+ I LI8j& rLP +RTw I + A, CeRe I-zatRe,i-qP4RTW )- R 61jc..aiQ +L P + TT:~p 8 A77B
+ q s~ rL P +-TA __fjJ" L 8A RGO, r+P +RT BT
-,
9,GTwA 2 R 2C BizfFAL (2A"~+ 6e?)~,ra(A4 +2 82)J p LI (2 8 + Az)a3/2
15(2 .')3r+' 20(- -O 2 )A' + 163('28)aA A + 8 A,1 I
,,'',+ Al
(A + 302A +728(a8,")A, - +34B 0 1 A, A 2. ( 2 A 6 A 2 .,
____ ___ ___ ) , cB o A (1,; o
[ Lo~.aa23+z~o2aZZ~ + 68(8 (Aa?)- - , ., A JJ '
-I--~~~~~~~~~~~~ +O42~2A A 782,)~3+ ] ~[$~~32
A
0 3oa,.(2231 ) . + 2 1 2)A B A +_ 2) 4
+-a z3 + 2 . )A ( +* 8 ;A J' ,.
ta k ' 2. _
,ii0<~ ~ ~ ~~~~)7 10,.. , 5-,, (-,. 8, - + 210 o(- ,,z,).,A +,< 16 8 , ,,a:- A + 4,-8,..
6-8 28 It6 1- t2JJ -A2-+(,2 B. )zA +) 2s_ 13,2 A, + 3 4 + 2_!. e, , i t A
;!J-A,;B + j-G+" (A.+ B .t aIX/)A"A 'A" - BI r- A ( + .- IAA+~WA-) A{ At (2A>-"3+,- A2
Rh++ I 3 +Ca o Art + , +0,
B) " (A"?+ + 2T (2a~ (I+ 2B)A z +. a L B 1 6( B-IA)
1I (8& +B )B+ .. A, )/2. A, T X I( I +B a)A + A A4
1 .2TO
COO
CD -
+ O oc s--
+ 0.I
co +l .4.o4 -,
k,, I- I-- C 1 -'
In ..- - - I -
f". +q -.- W...
4. L-- 0g tt c. 00 - , &.
0 + 000. +
co C: -- co - t
43 - + +, co- + 0~L..0-- 0 'o ", r-' -
'U r
0. + , o1 '
01+ .. o + . I 4
-+ go~''- +
L-0 - C14"
+r.4 1 co.0 0
+ I. o do 0 XI
o -- + 40 4- 4 -;
CO 0. a .411
0l 0-
d. 0 OD N~4
+ co N-v Q+ 0 cI..
+ 4-+ ~ -L .4
r'.0 0 +-0 ..I-V - If 4~
-. r'- co-_
+ !~ - 0- co7
-41 11~. IM - cT~ y~-- ~ -IN +4 +, - CO'40
+ . ...n .00A14 (
4No
0- co" .4 16 -u - P~0
to + cCD a" 0 0 - -4
m4 14 + N
+v c2 CIo Nm
Al At F9~~~[ 4s( 81- -BV)' + 25-O(1'+ 8) A ' + 32 4(8 2,+ B')' A,' 4) + r8 A A
[ 2L A, At (2 AL + S-( 5(31)) A A z ( .8 1B/ Laa + A.5
+o B~I +BZ+ ( 728(BA4. 8 P]8A'(aA, + 3(B; 5"))
F_______________________B, BFA, + 8 A i-+l.(6 8(B +B" A .. 2+~~~s LAi~B+~, 2 A
1 0 1 B2 8 2f A + 1 8 + A A +4T- _ _ _ _ _ _ _ A 4B -a ]z + 3 (, 8,24 )8D
2+Az )' /2
[0 1 W , -8,4. .) 8", ++ -.+A?12~±r'f~ 1 1 hA2fF
+ A,' + 2G T, p tan -1 ( 17+'~ +T~s~ R ~ A +z +,~ 2. +8- ' T + 13') (A' +
If the residual Equation 3.22 is multiplied by the second weighting function, 2Mequation that can be handled exactly as was Equation 3.23. Noting again that theas in the previous case with details of the integrations given in Appendix A and IThis second momentum equation is given by
_ I , B, _ , I _ _
-,, 8 [] I ta ,o, ,~ _,8( ) . ,irF - X L(lz "a 3/ (82 ,z)1 2 Bi + (ZY 28,L(1-;8 Fi
-B (, -2/ \(BS8Y' / (.)O 8)22 14-2 _*2 2F 8z Tj ) 2F + B2 S1
82 +, , 3 -, JA , ) , ]0.7,7)+-1120. r tan
20.F
)'-a 5_-2063' +- 8')" A' + "30o2+(8'+ B'Y) A,++1 S )31 JT 8 , ) A 16 + "384- ,'
+ +20. (37, + AAa + 88-&-AA5]]
#i ' )'B' + 0 L 16 , (B'2 (B, + BDI A' ))5-'/B2 + 13'a ( , + A -a ) m
-
('.4-2 AZ ( 13,1B(+B la )Me+8)A -+ 4-B A 9+ e,T [2 (B81-B)/a th [--
+ + -- +J(;w
~by the second weighting function, 7 -f tx,iq@e', one gets an integral'Equation 3.23. Noting again that the integrals are analytically.obtainedhintegrations given in Appendix A and B, one can thus solve for a.&
-- J AB4.- .-
61t (8(!8 B 1) + 8,1) ( 2 +A') F/2
B, + 8 -1 4 -r a., [ B, I- / B= )
, 2 Fir B, [ 8'2) (A") + + S' + .2
B2C'(2.7rL) B- 9a t A + r A
_____ IT i_____
8 B (B+5+B) (r 8' 8247+ .+ 8-. ) 1,L 2. B (a
d ytescn egtngfnto ,7,IAU * -oegt nitga
. .4 .
00 NO
4- ccJ ON co 4. 1
P, -I g
co~~ ~ .0c 11 0hN +to4 + 1 cco 4s - 0 o
4 - - 4.o -go Z ,, --
10 41 C Nc..a co + 41 N
-4. + -'
00c.4 . coc + A.
M 10w t +cC: -~4 I - +
'i;~ 4 . c
00 -4- coto- ON- W
~ c4o
c01+ ~ r 4"4Il_ -- ~-
10
O4 . 0 o ( + ( n+ 0 .-
j N +4to
0 *
~ C'.$ 4 -
j r ; 4 - C ccNj 44- 4jo "- t
ON 4.1 C:
V, ' 4 41 0-' ;-+ L c a . ,
+______ + +, -a -, a 2 ,
+4-r -r -8 ) T -r-
4- - , 8, - "
- 3 B -B ? 4 . -, 13 B .1- 12 . . . .
AA.,B 11 ir A
..(C) . AL., F3 3 1 Z a B, 32 + B . _B,_ . , . _, 2 .A ,, A _L
-x B, . (a8*+ a) + 2- B. 3 B2. ) +_
-_""-__
+TT- 7-T Bi O.Tw B " ' 7'.Ja2 -x (8>+ '£ A u 1;7-
± I 2% 3 2. l R tan'(OL,7) , -P B+ (2,A 2B B )
t I -az -3.B F )(B, - B2) - 82. 3 C) (B, 1 A )2 ' + 9-"0-
2- a, Bl a A a )-k 1 'B 8 A - AA.Id. ( 8? qi-2 ) (x I+ B, B)
QAA- -a I - AA- -+a"A
4- -iBI I +w S. -a R ,AI ,-zR9-A BA +-SL + TrL 8* (A +A
I ~ c GAL, LP + RTw I IIRG,+v~-+ RT C, ± f-2 g RG, A, AL., 4-7G*+R
I 1cJ., r 1 P+ RTw I C R F . I_ ____
3-x R G, +rl P +R TwB4 Ir J PL (A2t+tBBA) (ZAt+ 4B~"
+ ofA,A A- + G,'~ A'~B-4 2 i~ &Tw A,A,--t B" 75 A, 4 A'z Bz 2 ± G A 24 CA+ P-'' 8 (2 Az + B. +(8na
9C~r2 r A(2A4*3(8.Z+B%?) AlF 4,I(~B). - 0(S.4+ 82') A, + 8 A4,
+a-1U (Bl% J) 2A2 + I 8(a-4B)A4 + 4-8A, AA,] 9+ BL 4)+\4
1-2 B7+ a)A +-38 -A, + . A, (2 A~t -(~.B) A, r i
0 -16usB;)i 34.t l +~t +' a"B)BB.~/ [1 M
A-L
B, I 2 a ., I )., 4 - B, _
(2_____ 81 4-1 8 __Faxl 8, 67___ 2811+_
77x Bl,+-2. B + + --, (81- + 2---
3 r. ' a ( -2 Ai- d-13,S1- __, _-' 2, B, - 2.- -q B, \-
. ,a x, (8 .,.-a, [ , iJ -,x B, (8,2+B
,- .. ,.. a, , B, I + 4 - I + 2 .Ai
pq -j + +* T71B2
1 irFii BB. B, I I 6Bc,,., I
_ _ _ __,_ _ _ _ _ 6_ , I _ _ _ _'-c - RG,+r, P+RTw BI(A? + .a± - - $ z G A x L -- R,,PRw B( B) /
:, ReA , ax Pe_+_P+R wa I I I ___ __
- BkP+ T 8-x Rg,+r,.P+ RT,, B%(A +.B-)'/a
: I _ _ _ _ _]
~A?+8- i) (2 ,AI+ , + 8 )I (" e ,,A ,>- B8,- B ) + O,ThA ,_ ('A + ) j
~1" lo6',+ l B,+ ) a) '
. "8(1 B )4 Air AI 21(B "')+2 5"BO(S +) .)2A,) +,
RB 3 _1( B+*B + - I 8(8a, A -B)-(.,A aI - A28 / z B. B
2- 52( 8c.x11. 21 2 rJ,, C -
3 ) Z- C L
. -e 4.99
+0 - 0 (V
9.. +- ...- P ~+v .. .4.
.0 c9 0 r- Cco 4 + + 0 c.
+9 -v 1.0o 0
- ~ 0 .co0 ~ 0 .--+ +l .L 9. + L..
0v 0 -cV + +... +. 499 _ _ -- 9
+-f + co m 1..0.,1.; -9.o (V ft 4- 0 - +
CO1 . 9 .. f4 to ca
+ a4
. f0~~I 0 . 4 0 1Cc co* (V
0-.99 0. 9,4co +cn
0o Co4 .j r - 0 0 *ft -, ± ILI 49D 4.V
00 141L O 4o 4_:0.c Z -
09 +.+ r --- c C
+t 0 l + o
to 10 4K # 00 +. r'- - 0
0~~ + +l +~+ 00 9-~
co co tN 0 .1 0 0< 4
+ (n -V 0.N
0- 0. +Q + c c
ft I' o1, < (f 0 C9.cc
*I.~0 d + V
9 99
o. + +l cri f3 r o
'K + co 15 CO + 9..4; 1... 4 -- . .3
40 co: t, coc *9l 0
( - .c -oI. ~r - +l ( V 4 ". t 4 C O f 9 ( .
N~C 99' <.0 0 ri C0 -1 . - 0 %(
II+ 4 999 +9'. 0 ca C, '
0 0 co9
4t. ft.ff co +4 + r.-. -0- L 0, cc -0 ..
N~ ..( do ...
(91 CJ C ... J v 4o(I 9
co - . ':~. j
+ +
+ + 0. +0 l4.~c ++ .. ~,0~ 0 4
9(o 0 1
~c 0"V~< <i . " .I0. j-4 PON ,o c o0- 1
+i +c + + + *o 03 4. 4. + + '
09 0- 0,- ~ 9 0. << 9 + to 4 .- ,<-9>s-!-<~ .-
-~l~3A.,(2 A + 3
+. 20 184 +) Iq6(, +B)A'+B? 1A.) 71 1 rO 0 ( a+
D"+ (8A 8. 3 B AFr(B+- 1 1+A-B.9 I( )'" IB,( 8+)+A 4 3024-(
+ +8Wj(l' 2), 2 A6'( A7% A ) + + B)+B2+t - 0 + ,4 +L2'' tA22 (T+2+ 3, 2. +Bf BBf ' +' - BA 2 -IA7
) ~~~~2 + T -, '
++4Z + C ",A, (A, BB) +-L C vYGA 2 (A2. 8c8a e4B8,, ' t' -,
+ (B 2 )(Aa+& 2+B 2 ) -- L (B4+B )3/1ta (B,+Bt)(A+Bi+A +) + Tw -
±I2\'2. f2Cr+GA, A. e2, A.2J, 8'l) ('. B A})BI)( A+(A2 2 B )(A7-A1+2"-+ (A" -13.)(A+ A:
9, At T el A -z T 2RC r0c 8 A)(2 A'14 682) A'l A,( 60 + + 0A,B8 8 At)4 +A+2B) -+ (Ai+2B') p 16 BL,63(2B+ T 6+A6+132 B2z + AI) " /2
A,50 B 1 AA(2
A2 + ( B)A + 3 0 24.( 821+ 821)'4A++I-1 8(B +81)A ! + 3 84 4- 2. 4, -- 2 B )(l. i
+ "o + ( A (8168 ae2"))A, 4 - - +8 A -Al A-
10 1 0 (2 8 *" N " + " 2___42A
- 2 ) A + 728(28 )Al + 384- A' +-]-- )22 +( 6 (3))2A2
A2. 4]TV] Cr ,A. CToIA2 [ F2 - 2)).. oj.~- I
z[F , (2 B2)82
~22,
-A4 + 317 28(822A 26 +384 AaZ4- + el A(A+38.)
F+
[94s ) iB+B .)- +-' -SOB+8)A '(-+BI' (, B)A B.-
Aa ,, + 1 ATw
10 alB+ Ai81+8) A+ 2 ,
&' B+~a+ 2 'a+B A z 411 )-aA1+128 :+-8 4
A,,A +~,A __ __ __ __ _
Tr A (2A'4-6BI) A A,(6OB OA,8B- 8A )
+'1 " a" A.%+ 8 a -Az
LA +OC(YJ S,' 2 A? "
(AI-+ tTB=2 7 rA4+ 2 37 tat,- A,
~~~f-~~~4 r788(+ 51)A, 8,42 j~ & (B~~2
r-l la1/ + e~ ~22 ~ 60 A x(OB -'A.~B a A~
4AAa + 2 J-2A~ J Gs' ~ i A' L
2 a.A"A% 1 A' 0(g
A,2iA 6 z++ ,8 t
'14"6 2%A
Ci rV cu co
zV 3,1c I-r
L.J >~ 00 0
I, C)* 49 'C 1 41
+ .0 40 9
0. 4- E j 41 (4me (0 O C
4r cd 0.-II
S_ 0
C, (14 00 -- -- n
C; IZE 0 0-
%_ It 4J 4 4y< +
41' 0 0 14
4'
L4 +o 41Z01
d I40 4.4
.nM. 0 1(01
J"
+4 IIC'~a 0. 44. 4 O 4 .iIS_ 00 101,00C 0
06 14 a' 04 o
-1 4 C 0 .
'4 C 4 - w wcdi co +~ 0
rr 41 < M
S_ 14 4.
0 -D -0 tNj d di-A c -I>. +0) + u M -0)
ca wI c L.6 +
The following additional information is necessary to perform the analysis similar
ai at-r(1, - e + ) t , 4
-- 2- , Aq AA.X: (Aae +_. eA + o -2. (A q +! - , )A?- e-
Formation of the residual equation for the Energy Equation 3.6 is analogous to th'residual. Analogously, 2 Energy Equations are then formed by multiplyin2 the resrespective weighting functions,(21N,)s, 1 e-A~l and (_)2 . . -ai. . Asdetails of the entire expansion are bypassed. After one obtains the residual eqiover y, exactly as in the Momentum Equations, these 2 Energy Equations become
IA C) IE),. Az 1 2. cc) AA 6162.c ~S. -P go.~'' 'T C- at CA+~/ -9 7= t (A2,+A2-y/ AT- t-a ,(p at A, -(AT,8 IA )' /
Q - -
- , 777Fs, -- 72 + -72IB=). tn (a;.;), n._.
8, ,I( A,a )]( ): +~ ~ ~~~~~I A,(AnB , 42F- A,} A-,/T ( -'t + 8 a, I-B' A, ;
+ r, c a , n A A, C AB , B,1L(A +AAN)"1 Vc a' AAy 1 + (A'A i)( A7FA+ B)
Ba~~ X A5 F1 2 Al,
23"
-~ P B. .t~ ~ E£*.& £ K-.-~~~s.
-ssiary to perform the analysis similar to the above on Equation 3.6.
c)Aj~~~A 2 -A-A )9 c A
;t+7ea f [3. 2 1
[3.27]
Energy Equation 3.6 is analogous to the formation of the Momentum Equationare then formed by multiplying the residual Energy Equation by its two
and (_/,r) Szq .- Azj . As this expansion becomes very lengthy,d. After one obtains the residual equations and performs the integrationss, these 2 Energy Equations become
A 8 AA I61a. 1 S' 6TW I• -SI
o ~, t (A'2+ACP T . + A oW! o, [a 4F ,) *, Allt A~.-n) 1 .Af+Bi, rt-(A?+AgYfa(A c )T c- X t A ,/a(?Ay
2 AA -z A
tan At , '218,a .j
__.9_ al C1 eII 4k1 a 8j I
T. A, , n A, A-At(82-+2 A,- )
-, + = -, 8 P a , A , i- ,,;~" (A 12_A'za)%/'a (A-'%. A zA + A2-+ Bz J i -r $~ J a - A.,-2A
aeso' 82" CPe 2A+ 'j-an(4+AtV A Xa+ )T A'2+82N X, +a (BI A,)] - a 2
8%2 : -
zt4,
4. C',
-'V ca~ - 1C)
+ L -0~ - ~
+ ~+
14 't * -0
ca4
CD + 4..(vi OD 4
0 + ff.-
A 04. V .4' -0.- )?
-04 e-
+ I!
I L. ca-C.+ 0-C ~
4i -'I-
.4- 0.
.4..~t 4.4 I, . 4
4 - 4 -'C - ~ Cr~ (J14~ji~-~ L~'+
ct + ~ 4 . go C; LO ex - 9
La to
1 .i.,SI, ___ B .c)g' :T,,,BCp4- A A C AG, , e) T,,l - p , _)
(Al +A)(B7-A-+x A2 A +BB)' 2 8-x A(A -- B
, 9% ., 0 c,2) 3 ' -
A- J I-G ,, + 2 . 1 i --1 B z +? -T
1o4 8 I, z$ 4 . z c)B ____ _., A+ .0.8, _____
+ (A t '4- ~ .a SJ +- +p, - , A2 -l +eL , A',
A, t.,u., AA ( B + A, .QTTz 8 C.
o'x ? (A,+a ,rJ C O'xA, x (A.-A c+?a- B A-)
+C ,eA.,--, - c , i, £2S~ c_,__ o c , ,
+ - + C p Q J., A
~-x (A~+A~+ B ) (?-A2+(2t-8~2AF7 2 382+ AL + ~ 6 cB Ax (AzS, +-A
- - ,, ~, ALT 1=+~+~ - p 0" CGAL,A-b_ r c) Ix B,- +, e~ A~a +_r)f r PO t6 2. At + 1'4 r
o-I
+7 -a, So 9z 4AT+ cp 19 AL3, A C__ __ _ p 0T 1.P2 1 C
__ x _____A__+ ___ A-L 1 P + . CSp
-24 (A1 - a
C) TTw 8.1 ~ ,B
a C' 19 At T'--- -a CpO A td c3x A,(A +B.2)s/ r-- ' )-x AN(A2+8")~ aii-U2-fAl a A,
__________ B j
A, Cp OdB1 A 3aL n , .f22 +-CAT) + ,-7TAAag(2A~~+B~I i FTF (AlA~B~ +t%/i x (A +~
A + + A1 -f1,, a, A.~ n', AL .12AtCA(2A X B'I (2ArB~ IT C x ('+-'+FT 7OrC''7 x Y 7
adM. A -a, 91 2 al 2
df -x B 1%(aA'14 132 , I Wir- ~ d Ix A~ 4' +~B
a82 A' +l S 4- _____
c__________ Az C eAL, A,-61A B (~AA 2 +A!X+ arJ I T- Ix C2'+B 2 r
C)f 0, -i A AA8 -l CPG ()AA) 4-- CS , A
2x c.1 (A 4 -
CT g Cp 91ApC, AtE~ 4- AtG ~TP~ PR p~ 4W p~r Afr (2)A4-~" itJ + A -282)
At, a ~ T 1T 8_2aLAl____6.
+' 9 1i C p 9,r A Sirc x~j AIS R,rrC p 1 A I o- ,B
,, T T I( 8) + TTVB+B)2 ~-7A( 2 ( A a A.a rjw7A
GI lt 2 Cp e % C j 9,97 + -x 8 01A~t ,1 _ T
C; Cd
,'-,
C6 co ca
cc -1c 9x cc 0. o I 1"I I f-.I: I -
0' 4'I 4 C, 6 1cc
_ - - _. .Cl it -. " lz
CC "0
bIn 1
' II'
CI 0
, Ip
4140
cc+ .I-, CC
+.' co " cc '3 4.+ C
4l CIO.
vc CI < le0 06
+ + + + Icc W
Cd c Cd
Id rd +1 4 4 + 4 ic
*~ ~ + -. *cI-I - -
LJ J I , [ 11J-i L-
0- ++ ~ 0: tI +t-+ cc 0. ~
0.- cca 0o. 0 0. C,
cc cc cc cc4 '6~cc ~-4 cc 4+ 0.o + ~ :IT
CD+~ - 0 D c cc +a .4 0 c cc - .cc cc cc cc +14 CD. ~ -.0 * tC l xa jcb. C IV. 4 CD
4 Oj0 'i-j, -- I 4. *
0q C, C',
- --- )AL1 62,~& 1_ ' 1IB ~ ,, 1(A )'.t Re, +vPR , L y~ 2 I IApB
TT~ FtA~- RG+l+Tw AA, Az~ 1)/.tas(~+ll
2. cAL, R-fL2- , G'- ~ . _ _ __ _ _ A,
- ''t RG,+riPiRTw LJ-2 AaA Al(AL+ B) 1 /2 (
ir &LLct RG r -PRT w ra Az, F?. A +.cIt
)AAL, YiP RO+ RTwA (AI'-8t's) -z 0 tC)L R ,PRv Lr a n
___ R-fla,6 r -I B, A -~ir 4:r - R 1,+4-rL P +R Twk 42- ~ A2 Az (A"+ P,)"' a A2L + BJ)" t
2~ ~ n% ________ 1 8a~ (&.-L2 A,
frJiO RAlRT 4-2l \1 A. Aa (A +B 13 4 A i A ZY
l~e~a~ R R_ G+ + qP A-~S,
+'" R C, - . 8- IL +~ RC,8 -AtA. ir -x R, + rP + RTw ~ ~ ta. RO, +?P+L t A~AI2
±1~ ~~~ A., ~4R ~~,} I6
4LF + 9 2- AA., A
__ __ L-R,+ ~ -Aewc , + rL P2~A -j RTw+ %
+ T-axtk A(A) A', R62 '1 aA t R E,8 A
2.f- c)-A A , [ tP+RT A, __ I l A__3
tan' Ba + t A' (Al +a A') a
Ce.u~-. A(A + +
- -~~= e~±,+±.[ ~ t&'( ~ )I87) 7?-- ~-t~ B
sls 4G ( ' TI25).At+ %'/- A
tan-_ A 6AJ- ~L P A1 +Tw )Ql75- A2 (A-B-)%/ c tn R 6'~ RO+v P + T ATB 1 +
~aJ'~B )n-_ _ _ _ A, V
AJ(- L ++3j-2 'k)7
AA 2
an ez 2 +_____(__+_8'_ t /' 1-I '
7T RG, +LP +R TI- Wt A82
Kita '~ - -~ ____ ___C) AL, ,iF + AT,,, __ __ __
,tank 2 + 2 '( AP0,)] Ae,+Rw8228
(2 CA, Az (A'%I(A~4A + Bt A
-I~~ _______
A'rA, B?'r IA -A1 O LIAx+ A) t 7
18 C
2' V~All
4- 4.14 .
- a.+'r*q
0
-0 + 'o40-
-~ ~ ~ L Le 14L1C. -- S .--
Q'~ q- 10 14 +I t co' -* Do b- O_- of- + ~ ~ , I
+S I I &Z 4 I
"e 1 + vO 4o 4.C
0. ca
-C- +
+- Ile C, + co~.I:+. op L CQI
-,, 1 j
L' C+ 10'+.j -< o , l
co 4j Ci .. XH iC, fC, c;- 0 4-t tocC1 Vo,~
4u u 0K
+ I + - - _ I +I I ' +
+ (AI 11(a~4) ] 2 1 1 /* % n~I(. AI A) A " A (AL-
I- CI (-r? A - ~ ( Z t 8 T A '4 1 "i+r ZAiBr C) -r2:
72 A(2-i-- +C A~ -' A%+ ±' ] x*
Rra- + T776, A* ' 7zT,
+-- *C ce% - A -x It x -I ) I3' +A- ~ 6z-B B. 91
ir - 7 B 2 4 A% AA4( 2 4I
Ai~2 ~ n _)~ -a, 3 8 , &Z2+C p A A , +___ XCa,&tE)sJ
- i--cef ~ A ( AI'L cp je- B, ~ ~~ ~ ~ cBA
Z c2 3 A,~ C) Bi~66 A A? a + ~ d 2 A
-x (,A7 +.3j A t+61 rC t ' lS ?. Re~I M - x A(A + A'z +
-p at SOCI C 2Al, 8
8t B - __1____A,
+a +Aa(,AV t a)
(A +- A-)'- CAG: 1 -- A, (AX-A 2 51
A Tr
IL B417- (AllB~ -BZ____(A_4 8 )'_ i _
6 A1 + 38 A
+, Sx (2c + 8 )2
pO~~A 7F -a pe~2A,____(A3x4 A' )3~+~ (T4i (Ao +~ A) 1AAt At ) (
9,eA.~ A, C. G 17 aw,(7
A (ZB A. + 5, 7&1r IA + B 2aS
-A ____tan' " + 482 A, Gi Ak A.B ~2 -~7a7 8_Ax +A- 7 XcLeAA' + S(ak-C) -x (A At+8Y
+ __)_t AA4 ,' T & A. 8a 2. St 2. C1-8h- A
2 1 ir Air oxA
-L i- 418.A.kfl~~24
ve _. 4I, c
cle CCOee 0.<r-v
01 1 rj- I
Ow i0- III -
+ I ic cI0 Z ' <0 0 L:' -o
-i Co, to
4- Q F- I < -1 C 4 -C
1: 0 Irl 4.
<0 I (d. W
41L 0 4) 4
-~ ~ ~ C K..- +4.CJ
a. 4 . cca 0 0.'- ' a:a+ ~ 03a; e -E ;: vr' C;: i 0. F-4-.~ ~ c 1-4. + cc -+ 0
-k!cc O I V , .cc + to +. (D+ +. to + Ix .0. Ca: c. c - :
-: 014 L..I- 4. Ctu ~ ~ ~ C ' 4 0 ~ ~- ZT cc- cc e ) c t- c
1- + + I + I I + + lb
x~OZ
.I a T 91 fLP+ RTw +. - C k)A S' 2 A, + A,R,+t e+RTw 4&. x. Re,-rLP+RTw AZ (A'+A )',. + 1-.
I ,u.,. T, h.P 4- RTw - . 8 C) 0 . .P Sigt [A A,+. -4 ROe AL ' t Rw +,)P+RT + ^=:
B8 A, 2. [AL R a B,)+ -a - -x At,,,
au Y tP+ ATw a)a 82 2)" &,aA. ~r__at RA,{(+ P+ w A (A1+82 6V' -- A 1 +,rt P + R AAT
A'%RrP+RTw A +A'7+B7)0t hh at R-x O,+,tp+ RTw A
+(( B; ' (A + a;)B'/ - -V e, ' a x Re,+ TP .' , A,
aA4.L, ,P+ RTw B. - S z 2 AL AR., R , . At
-- ~~ 2 az. ta=k dA R-a,62 . ,T A.A 1+ ,+ + T,, r. WA . a)h'
AZAZ +, BI u. --7 1 6% R,,, G, ,.,,, +_ R T, Aa 4 ...A..I r ' t PB-I- +RTw + 'j.S- 2_za)" Re, qP+8Jz At(,+
+ e a R9,+rtP+RTw , o't).v/-,. 7x RG,+YP+F~w
84 A, A. -a2M- 7-, ) - - e 1 LAL-' zA. -
__ __ __ + _ _ _ I) - L P +_ AA,, L-' 2
~JF~RG,+qP+PRTw A,-LA-z'+ 0± 1 ~' 3T RO,+PLP+RTW A7-(A-&+A")
GI rL P +TW 8 ALI R So&- A,
+x RGi--PRX RG,(AP+RA4-FA
AThereby, the objective stated in the beginning of thi 's Section has been achieved.Equations 3.24, 3.25, 3.28 and 3.29. These are still partial differential equationindependent variables. The independent variable x will be eliminated in the next S
2-7,.
'A
A SO , -Z+ A t? R' + p k.8"-x' R x , -t- -P+RT A?-A+A')" 7 CJfr k''a-x R, ,- P -RTw
. 2. t, du, RellZ . -'A, ,\ B, Ai(A+y7i- -F C) t A at+ P+RTV. ALIAx+ATFE (A'+ AlFL)
191 A - - +~~rA -A 7z))+~ ~ ~ C -,6z Ft4 9' ''J , n.+ R Tw A'z (A,- + ' (A, + AI) '
.4"2
- -. e , RmLa [9 + A~ __
Re,+- tPP A .RTw 1A (AA%) A" (A'-$1- +,,,.IL
n't? 3 -Z1 L C-L a,8 1) BI' 49 AL_ --,6 ta ' B ' (A-+___ -I A
2j OL eA~~M a-" )-a GA 1(B )+ 81A+~I/CLh At.)
S-"x R,1,-lP+-RTw F2- (AI+B .)2_____a_____i,8S' A t h 7 du. RS
+ B)3 ± 2- 8% RO,-i.P+RTw A.(A(+A/7 A It 1
I , . A A( RAAa A , + I +O 1dA Rz
+ 317 6-X RGJ*I-y1 P+RTW AW@ - Aat 27jfrAd t Re, + P*T +1-F I " -d" ,+ r P + RTw A2 (A4+ ,'" "T- R 0"-' + " r-'P *e, r ,RTw
he beginning of this Section has been achieved. The resulting equations are19. These are still partial differential equations but with only twoIndent variable x will be eliminated in the next Section.
27.2
3.2 Method of Lines
The Method of Lines39 is a technique commonly usedto solve partial differential equations on general-purposeanalog computers. When two independent variables exist,say x and t as in the present problem, one independentvariable is discretized and the other is permitted to re-main continuous, usually with respect to time t. Thepartial differential equation is thereby converted to asystem of ordinary differential equations, General-purposeanalog computers are intrinsicaily capable of handling onlyordinary differential equations.
The usual technique of solving partial differentialequations on a digital computer involves the approximatingof derivatives with respect to all independent variablesby finite difference expressions. The resulting algebraicrelationships are then solved by matrix methods at the gridpoints of the discretized region of interest. Another possi-bility for the digital computer solution is to convert to asystem of ordinary differential equations as in the Methodof Lines and then to solve the system by standard techniquessuch as Runga-Kutta or Predictor-Corrector methods availablefor ordinary differential equations. This is the approachthat will be used in the present investigation. The estab-lished numerical techniques witn automatic step-size controland automatic error control for solving ordinary differentialequations can be used. Moreover, Smith and Clutter "0 solvedseveral boundary layer problems by the method in conjunctionwith high-speed digital computers, Koob and Abbott' solvedthe unsteady incompressible boundary layer equations withoutpressure gradients by use of the Method of Lines and themethod of weighted residuals. Hicks and Wei 2 solved theunsteady one-dimensional heat diffusion equation by theMethod of Lines. This approach yields results with lesscomputer time than the explicit or implicit finite-differencemethods.
The basic assumption of the Method of Lines is that con-tinuous functions such as B1 , B2, A,, and A2 and one of theirderivatives such as
3B DB A @A, , x and----L can be approximated in terms of
several new functions such as B1(xj), Bl(x2),..B 1 (Xn),
B2 (x ),'''B 2(xn), A1 (x,),'"*A1 (x n) A (x,),'"A 2 (xn)
28
B (x )-B (x) B (x)-B(x n-) B (x )- (x)
x-x x -x x-x2" 1 n n-2
B (x )-B (xn) A (x )-A (x ) A (x )-A (Xni2 n 2 n- 2 1 n n-
X n- Xn x-x -n- xn_
A (x )-A 2 (x 1 A (x)-a (x2 -x n Xn-i
Here, the field of interest is assumed to be betweenx, and x . The derivative of continuous functions is written!n a backward difference form. If the information above isincorporated into Equations 3.23, 3.24, 3.28, and 3.29, asystem of ordinary differential equations (4(n-l)) resultswith time as the only independent variable. The locations xito x need not be at equal intervals. The solution of theresuiting equations will be discussed in the next section.
3.3 Boundary Layer Parameters
The objective of the present investigation is to obtainthe solution of unsteady compressible boundary layer equationsstated in Section 2. The end results are in the form ofvelocity and temperature profiles as a function of the in-dependent variables x and t. Once these profiles are deter-mined, any other parameter of the boundary layer can beeasily obtained. The derivation or evaluation of theseparameters is given below:
Heat Transfer: The heat transfer at the wall by con-duction is given--by
Q - (KDT) = h(T -T1 ) (3.30)Qw -(Ty y =O0W
Instead of introducing a model for thermal conductivityK which is a function of temperature, another form ofEquation 3.30 may prove to be useful.
Q = P _ 2.-1 (3.31)w Pr Y=0
29
The following final form is obtained by use ofEquation 3.11.
Qw P r .... -,(&A, + 2A2 ) (3.32)
Pr poV (3.3
The convective heat transfer coefficient as definedby Equation 3.30 becomes
h wp .61A1 +62A2)
Pr p0 Vi
0or Nuw _2D Pw(or po A +6 A2
Qw 2 pwPwor St=- C u (6 A +62A2 ) (3.33)
0*
Shear Stress: Assuming a Newtonian fluid, the shearstress at the wall in Cartesian coordinates is
Tw (Pau) (3.34)
a y=O
This can be written in terms of new variables as
T = I (JWBP +QB ) (3.35)w 11 22
Finally, tile dimensionless skin friction coefficientbecomes
- 411 pw w wcf _P____ p (Q B1+'22B2) (3.36)
30
A-
Displacement thickness (6d): The boundary layer di.s-placement thickness is defined as -
(1- Pu )dy (3.37)d j (I puI
0
Equation 3.37 becomes, in terms of variables. in thepresent analysis,
I [p(RO. RTw +n) - u_ -L dy (3.38)d op p P1 ul
0
The following final form was obtained after substituting Equation3.11 and integrating analytically:
poRO 1 ) RTw
d : -- [ - ~+ ] + P ( T +!) 6
d 1 2
- (6- I ) + Q (6 I )] (3.39)p1 B/ 2 IT B2 /T1 2
The displacement thickness indicates the distance bywhich the external streamlines are shifted outward owing tothe formation of boundary layer.
Momentum thickness (6m):
CO
6m = pu - u_) dy
0
: PO (u [U_ )2] dy (340)
0
31
This parameter is useful in the determination oflaminar-turbulent transition and also indicates a measure ofloss of momentum in the boundary layer.
Energy-Dissipation thickness (6ed):
p. u I u_) dy6ed pI ul ul
0
00.
0
The parameter above indicates a loss of mechanicalenergy occurring in the boundary layer.
Enthalpy thickness (6h):
= J -- ( - I) dy
h pI U "1 TI
0
0~~~~Z . 42)+(W I -
0
Velocity thickness (6u):CO
6u (1 - ) dy (3.43)I U 1
32
This integral cannot be evaluated unless y is written
in terms of y by Equation 3.1. That is,,
J Tw + n) dy
p
A 7 /T
A2 Y 2 RTw
+ 6 (Y erf (A2y) + e- 2 -)] + P T + n) y (3.44)A2 V/ 0 p
Even though the integrals in Equations 3.40, 3.41, 3.42,and 3.43 can be evaluated analytically, this is not attemptedhere because of the tedious derivations involved. However,one can easily evaluate these integrals by Simpson's rule ofintegration.
4. NUMERICAL ANALYSIS
The unsteady compressible boundary layer problem statedin Section 2 contains a system of nonlinear parabolic partialdifferential equations with three dependent variables (u, v, andT) and three independent variables (x, y and t). The introduc-tion of the stream function ' eliminated one of the dependentvariables, namely v from the unknown list. The pressure andthe free-stream velocity are to be provided either from experi-ments or from unsteady core flow analysis to ciose the systemof equations. The variable, density, can be expressed interms of pressure and temperature by means of an equa*ion ofstate. The viscosity is considered as a function of tempera-ture.
The elimination of the dependent variable v introduced anew dependent variable p to replace the original dependentvariables u and v. The temperature T is replaced by a new de-pendent variable 6. The dependent variables p and 0 are stilla function of three independent variables x, y, and t. Themethod of weighted residuals, in particular, the Galerkinmet'~od, introduced the dependent variables Bi and B2 ,-o replace
San'd A2 and A2 to replace 0.
33
These dependent variables contain only the independentvariables x and t (Equations 3.23, 3.24, 3.28 and 3.29.The Method of Lines made possible the transformation ofthese partial differential equations into ordinary differ-ential equations with time as the only independent variable.However, this transformation is achieved only at the expenseof introducing additional unknown dependent variables, aset of B1 's, B2 's, A,'s, and A2 's. This set is dependentupon the number of discretized x stations. In a typicalexample considered in Section 5, eleven x stations are chosen.Since a boundary initial-value problem has been transformed to aninitial-value problem and thus requires specification of B1 ,B, A , and A, at the first station as a function of time,46 ordinary differential equations [4 times (11-1)] must besolved for that typical example to obtain the engineeringaccuracy. The number of equations will increase further ifone increases either discretized x stations or the number ofterms in the assumed approximate solution forms (Equation 3.11).The application of Method of Lines as explained in Section 3.2on Equations 3.23, 3.24, 3.28, and 3.29 yields the followingequations:
B1 (x2) = fl, 2 (B1(x2 ),B2(x2 ),A1 (x2),A2(x2))
62(x2) = f2 ,2 (B1 (x2 ),B2 (x2 ),A,(x 2 ),A2 (x2))
Al(x 2 ) = gl, 2 (Bl(x 2 ),B2(x2 ),Al(x 2),A2 (x2))
A2 (x2) = g2 ,2(B (x2 ),B2 (x2 ),A (x2 ) ,A2 (x2 ))
. .. a . . . . . . . . . . . . .
J(Xn = f n(Bl(Xn),B (X ),Al(Xn),A ( n)r ~C 1(Xn) = fC1,n(Bll(Xn) 'B (xn) 'Al(n) 'A 2 (xn ))
L2 (Xn ) 20f (B (xn ),B2 (xn),A1 (xn ),A2 (x n))
A1 (x n) = g ,n (B 1 (x n ),B2 (x n ),A 1 (x n ),A2 (x n))A2 (xn ) = g2 n(B 1 (Xn),B2 (xn),A 1 (x ),A2 (xn)) (4.1)
Where the dot denotes differentiation with respect to time, t.
34
The functions f's and g's are the same as the right sidesof Equations 3.23, 3.24, 3.28, and 3.29 with the exceptionof discretization of B's and A's and their derivatives.
The system of equations cited above may be solvedeither by standard Runga-Kutta techniques or by Predictor-Corrector methods. Since the functions in Equations 4.1 orin Equations 3.23, 3.24, 3.28, and 3.29 are complex and in-volve many tedious computations, some computer time may besaved by use of Predictor-Corrector methods. However, thesepredictor-corrector methods are not self-starting. Thus,initialization by ,+her techniques such as the Runga-Kuttamethods is desirabic. Since the available time is limitedand since, in the author's experience, the Runga-Kutta methodsare more stable and more easily workable, the standard fourthorder Runga-Kutta scheme is used for the entire investigation.
A fifth order Runga-Kutta integration scheme was alsoconsidered to improve the accuracy. The functions must becomputed six times instead of four times in a fourth-orderscheme for each time step. The additional computationsintroduce more errors. Moreover, Milne4 3 states that thepursuit of greater accuracy by derivation of formulae ofhigher order is a "losing game." The formulae rapidly be-come formidable complexity, and the large number of sub-stitutions for each time step increases computer time sig-nificantly for a slight increase in accuracy. Therefore,the fifth order Runga-Kutta method is not used ultimately.
Since the growth of the boundary layer is faster nearthe leading edge and slower downstream, the introductionof a variable step size for Ax is sometimes convenient.This can be accomplished easily without variable step sizesin numerical computations by means of the following trans-formation:
Let x = /i (4.2)
where L is the length on the plate up to the end of the flowfield under investigation. Since the variable x/L variesfrom 0 to 1, the equal step size of 0.1 can be obtained forx/L of 0, 0.01, 0.04, 0.09, 0.16, 0.25, 0.36, 0.49, 0.64,0.81, and 1.0 (ll Stations). Thus one can obtain eightstations for the first half of the flow field and threefor the remaining portion. This is achieved without avariable step size in the actual computer program. A fewnumerical examples will be considered in Section 5.
35
5. SAMPLE PROBLEMS
Not much literature is available on unsteady boundarylayers even though hundreds of investigators, throughoutthe world, are working on boundary layers. This is es-pecially true for compressible flow with pressure gradients.No known example exists that can be used directly as a testcase. Koob and Abbott4' solved an unsteady incompressibleboundary layer problem on a semi-infinite flat plate withzero pressure gradients. A similar problem was also con-sidered by Hall. This analysis is based on implicitfinite-difference method5. This example and the results
will be discussed in Section 5.1.
Numerous studies were conducted by Mirels 4 1,46147,48,49
for shock tube flow. Similarities are present between ashock tube flow and a gun tube flow. Indeed, these areidentical with the limiting case of bullet mass approachingzero. The growth of the boundary layer starts at the high-pressure end as well as at the base of the shock. Similar-ity profile were obtained by Mirels"S for steady boundarylayers. If the coordinate system is fixed to a shock wave,the flow in the boundary layer behind a shock wave may beinterpreted as if the flow is steady. However, if thecoordinate system is fixed to the shock tube, the sameboundary layer behind a shock wave may be interpreted as anunsteady compressible boundary layer flow problem."0 Thisis the only test known to the authors for an unsteady com-pressible boundary layer problem. This problem is discussedfurther in Section 5.2.
5.1 Rayleigh-Blasius Flow on a Flat Plate
No exact solution for unsteady compressible boundarylayers apparently exists that would provide a complete testof the present method. The compressibility, large pressuregradients, viscous dissipation, and,the gradients in timeas large as the gradients in space display quite significantdifferences in results from steady state in an ideal testcase. Since no such solution is known, the Rayleigh-Blasiusincompressible flow on a flat plate may reveal at leastsome unsteady boundary layer characteristics.
The present method may be tested by computation of asolution that should, at initial times, be identical withRayleigh's solution for an infinite flat plate started im-pulsively from rest and that should approach ultimatelyBlasius solution for a semi-infinite plate in a steadyuniform stream. The boundary conditions on an upstreamstation are required for all times. However, these are unavail-able for an impulsively started semi-infinite flat plate.
36
'I
Therefore, arbitrary conditions (identical with Halls 4 4 )will be imposed for the purpose of this investigation. TheRayleigh and Blasius problems are discussed below to com-plete the presentation of the example problem.
The governing equations of the Rayleigh problem, ormore commonly called "Stokes first problem" are
at 32U (5.1)
t<O: u=O for all y
t>O: u=u1 for y : ; u-O for y=O
Let n=-- and u f(n) (5.2)
If the definitions in Equation 5.2 are substituted intoEquation 5.1, the following ordinary differential equationand corresponding boundary conditions are obtained:
f" + 2nf' = 0
f(-) = 1, f(o) = 0 (5.3)
where prime denotes differentiation with respect to n.
The solution of Equation 5.3 is
u f(n) = erf (q) (5.4)U 1
Various parameters of boundary layers can be obtaired easilyby use of their definitions.
37
The governing equations of Blasius problem are
U_u + V a=0ax ay y
y=O: u=O, V=o
y2;: u u1 (5.5)
Let n =-Y and = Vxui g(n) (5.6)vxr-
Vul
The following definitions eliminate the second equationof Equation 5.5 from further analysis.
u = =ug'(n)
v '-' 3x = 'vu1 (rig' -g) (5.7)
If the definitions from Equations 5.6 and 5.7 are sub-stituted into Equation 5.5, the following equations are ob-tained:
29 11+ gg' = 0
g(o)=O, g' (0)=0, g' ( ) = 1 (5.8)
where prime denotes differentiation with respect to n.
No closed form solution exists except in series. Theresults by numerical integration are as follows:
38
TABLE I
0:4oF. gI Igb u~3~u g
0 0 0 0.332060.2 0.00664 0.06641 0.331990.4 0.02656 0.13277 0.331470.6 0.05974 0.19894 0.33008
0.8 0.10611 0.26471 0.327392.0 0.16551 0.32979 0.323011.2 0.23795 0.39378 0.316591.4 0.32298 0.45627 0.307871.6 0.42032 0.51U76 0.296671.8 0.52952 0.57477 0.282932.0 0.65003 0.62977 0.266752.2 0.78120 0.68132 0.248352.4 0.92230 0.72899 0.228092.6 1.07252 0.77246 0.206462.8 1.23099 0.81152 0.184013.0 1.39682 0.84605 0.161363.2 1.56911 0.87609 0.139133.4 1.74696 0.90177 0.017883.6 1.92954 0.92333 0.098093.8 2.11605 0.94112 0.080134.0 2.30576 0.95552 0.064244.2 2.49806 0.96696 0.050524.4 2.69238 0.97587 0.038974.6 2.88826 0.98269 0.029484.8 3.08534 0,98779 0.021875.0 3.28329 0.99155 0.015915.2 3.48189 0.99425 0.001345.4 3.68094 0.99616 0.007935.6 3.88031 0.99748 0.005435.8 4.07990 0.99838 0.003656.0 4.27926 0.99992 0.000227.2 5.47925 0.99996 MOM0i7.4 5.67924 0.99998 0.000077.6 5.87924 0.99999 0.00004
7.8 6.07923 1.00000 0.000028.0 6.27923 1.00000 0.000018.2 6. 47923 1. 00000 0.300018.4 6.679231. 00000 0.00000
39
Since the solutions to the Rayleigh and Blasius problemsare obtained, the test case can now be set up. The flowfield characteristics, and the initial and upstream con-ditions are set for compression of the present method withHall's implicit finite-difference results. The region ofinterest is l<_x 2. The free stream velocity (u,) is unity.The initial time is 0.5. The kinematic viscosity in thepresent analysis will be set to unity to match with Hall'snondimensional problem. The boundary layer developmentform the time t=0.5 is sought. The initial and boundaryconditions are as follows:
t=0.5, l.x:S2: u = erf (
x=l, 0.5x<lS.: u = erf Y2/v5-2
x=l , 1. O~x<00: u=erf ( Y )+[erf (2- )-ulg'(n]v- e-'.25(t-l)2 (5.9)
The function g,(n) is already listed above. The lastupstream condition represents an exponential variation fromthe Rayleigh to Blasius flow. This is an arbitrary conditionand not necessarily true for an impulsively started semi-infinite flat plate.
The initial and upstream conditions given above are notreadily acceptable in tie present computer program. Theseinitial and upstream conditions must be interpreted in termsof B,'s and B21 s. The definition of u in the present analysisis given in Equation 3.11. The Bi and B2 may be obtained bythe matching (collocation method) of this equation with Equation5.9. Since these equations are of transcendental type, onemust solve these equations for B, and B2 by the multidimen-sional Newton-Raphson or another similar technique. Sincethis approach consumes more computer time and also since theconditions at the wall are more important than in the flowfield for heat transfer or skin friction computations, thefollowing alternative is considered. If the slope of thevelocity component at the wall (3u/y at y=O) is matched,one of the unknowns can be expressed explicitly in terms ofthe other unknown. That is, B1 in terms of B2 or conversely:
40
/
2B 2Y2 2B2 _ 2
= B _e (5.10)
2u1Tu- - 2u(Q B1+ 2 B )
or B au - 02B2)/Q, (5.11)
=0
Note that y is the same as y due to the incompressible assump-tion.
The resulting unknown may be found by the matching of Equation5.10 with another point in the flow field neai the wall. Thisis accomplished by the one-dimensional Newton-Raphson tech-nique. The velocity gradient in Equations 5.10 and 5.11 canbe obtained from Equation 5.9. The results are discussed inSection 6.
5.2 Shock-Induced Boundary Layer Problem
This is a compressible boundary layer problem but with-out pressure gradients. The governing equations of steadycompressible boundary layers without pressure gradients wereanalyzed by Mirels 4 5 (similar to the analysis of the Blasiusproblem in Section 5.1) who integrated the resulting ordinarydifferential equations by numerical methods. The results
were tabulated with dimensionless wall to the free-streamvelocity as a parameter. The pertinent equations and a sampletable (uw/u, = 2) to interpret in terms of the variables inthe present investigation are as follows:
u (IyjRT b(yl 2'd = 2
(- 1 + 2ygRT
b
41
T (y+l) - - (y-l)Tb U-w (Y+I)- (Y"I 1% '"
T W+uw 1 2 r (n ) T T
--T_ = I+ + (T! - T ) s(n)1 2TiCP, w T T
Uw u 2r(o)) 2
U. -
UlU1
y
U T u dy (5.12)
j 2xvw
Note that the shock velocity and the wall velocity arethe same to interpret the results between steady (coordinatesystem fixed to the shock wave) and unsteady (coordinatesystem fixed to the wall) flows. The shock speed can beobtained by the relationship mentioned above. For the un-steady case, the coordinates of flow of interest may becomputed from the initial shock wave location, the shockspeed, and the elapsed time. Thus, initial and upstreamconditions may be obtained for the example under consider-ation.
42
TABLE II
Solution for w 2.0, Prandtl number aw = 0.72
) r -r s -s
0.0 2.0000 1.0191 0.8997 0.0000 1.0000 0.8512.1 1.8984 1.0091 .8922 .1479 .9151 .8452.2 1.7988 .9804 .8704 .2861 .8313 .8279.3 1.7029 .9356 .8356 .4068 .7498 .8004.4 1.6121 .8777 .7898 .5043 .6715 .7645
.5 1.5276 .8103 ..7356 .5758 .5972 .7217
.6 1.4503 .7367 .6755 .6209 .5274 .6739
.7 1.3304 .6601 .6122 .6412 .4625 .6227
.8 1.3183 .5834 .5480 .6398 .4029 .5697,9 1.2637 .5088 .4849 .6206 .3486 .5162
1.n 1.2164 .4382 .4244 .5878 .2996 .4636i.1 1.1759 .3728 .3676 .5455 .2558 .41271.2 1.1416 .3135 .3155 .4974 .2170 .36431.3 1.1129 .2606 .2683 .4465 .1829 .31891.4 1.0893 .2142 .2262 .3953 .1531 .2769
1.5 1.0699 .1742 .1892 .3456 .1274 .23861.6 1.0542 .1402 .1570 .2988 .1053 .20401.7 1.0417 .1116 .1293 .2556 .0864 .17311.8 1.0317 .0879 .1057 .2166 .0705 .14581,9 1.0239 .0686 .0858 .1819 .0572 .1219
2.0 1.0179 .0529 .0692 .1514 .0460 .10122.1 1.0133 .0404 .0554 .1250 .0368 .08332.2 1.0097 .0306 .0441 .1024 .0293 .06822.3 1.0071 .0229 .0348 .0832 .0231 .05532.4 1.0051 .0170 .0273 .0671 .0181 .0446
2.5 1.0036 .0124 .0213 .0538 0141 .03572.6 1.0026 .0090 .0165 .0427 .0109 .02842.7 1.0018 .0065 .0127 .0337 .0084 .02242.8 1.0012 .0046 .0097 .0264 .0064 .01752.9 1.O000 .0033 .0073 .0205 .0049 .0136
3.0 1.0006 .0023 .0055 .0158 .0037 .01053.1 1.0004 .0016 .0041 .0121 .0028 .00813.2 1.0003 .0011 .0031 .0092 .0020 .00613.3 1.0002 .0007 .0023 .0070 .0015 .00463.4 1.0001 .0005 .0017 .0052 .0011 .0035
3.5 1.0001 .0003 .0012 .0039 .0008 .00263.6 1.0000 .0002 .0009 .0029 .0006 .00193.7 1.0000 .0001 .0006 .0021 .0004 .00143.8 1.0000 .0001 .0005 .0015 .0003 .00103.9 1.0000 .0001 .0003 .0011 .0002 .0007
4.0 1.0000 .0000 .0002 .0008 .0001 .00054.1 .0002 .0006 .0001 .00044.2 .0001 .0004 .0001 .00034.3 .0001 .0003 .0001 .00024.4 .0001 .0002 .0000 .0001
4.5 .0000 .0001 .0000 .00014.6 .0000 .0001 .0000 .00014.7 .0000 .0001 .0000 .00004.8 .0000 .0000 .0000 .O0004.9
43
Similar to the determination of B1 and B2 in theRayleigh-Blasius flow problem in Section 5.1, not only B1and B. but also A, and A2 are to be determined to provideinitial and upstream conditions for the shock-inducedboundary layer problem. The entire procedure is the sameas before and need not be repeated here. The results arediscussed in Section 6.
6. RESULTS AND CONCLUSIONS
The overall gun tube heat transfer problem applicableto any weapon, ammunition, and firing schedule was analyzed.Toward this goal, the propellant gas convective heat trans-fer problem was divided into five problems: (1) generationof thermochemical properties for any given propellant, (2)transient inviscid compressible flow through the gun barrel(core flow), (3) transient viscous compressible flow on thebore surface (boundary layers), (4) unsteady heat diffusionthrough single or multilayer gun tube, and (5) unsteady freeconvection and radiation outside the gun tube. Limitedliterature and solutions to these problems were discussedin Sections I and 2.
With NASA-LEWIS "EC-71) thermochemical program (ChemicalEquilibrium Chemistry), the chemical composition and adia-batic flame temperature and thermodynamic properties werecomputed for M18 and IMR. The unsteady inviscid core flowproblem was solved by the method of characteristics. Thetransient two-dimensional heat diffusion through the guntube wall was analyzed by finite-element methods. The un-steady, two-dimensional, free convection and radiation aroundgun tubes with variable wall temperature was analyzed byexplicit finite-difference methods.
The transient, viscous, compressible flow on the boresurface with viscous dissipation and pressure gradients wasformulated in Section 2. The unsteady compressible boundarylayer on the bore surface is one of the most difficult prob-lems to analyze due to the limited State of the Art (almostnone) and also to the existence of laminar, transitionaland turbulent regions within the boundary layer. Since thedevelopment of the transitional region is not well understooa,the boundary layer is assumed to change suddenly from laminarto turbulent flow when the Reynolds' number, based on momen-tum thickness, reaches approximately 350. Similar assumptionwill be used for highly accelerated flow if laminarizationoccurs.
44
The governing equations of the unsteady compressibleboundary layers are a system of nonlinear, parabolic,partial differential equations with three independent
variables. The transverse coordinate was modified to ab-sorb the compressibility effect. The Stream function wasintroduced to satisfy the continuity equation and also toeliminate one of the dependent variables. The method ofweighted residuals was used to reduce by one the numberof independent variables (y). The approximate solution formwas chosen based upon the asymptotic solution of the steadydifferential equations for large values of the spacelikecoordinate. The error functions, consequently, occur inthe solution form for forced convective boundary layerproblems. The method of Galerkin was used as the errordistribution principle. All integrations across the boun-dary layer were performed analytically. The Method of Lineswas used to reduce the resulting partial differential equa-tions in two independent variables to an approximate set ofordinary differential equations. This procedure enables oneto solve for derivatives by the reduction of a matrix withelements of not more than the number of undetermined param-eters introduced into the solution form, and thus less com-puter time is required. With this analysis, density andtemperature variations are allowed not only across theboundary layer but also with time; also strong variationsof free-stream parameters are allowed with both axial loca-tion and time. The various boundary layer parameters werederived in Section 3.3.
The resulting equations were programmed for a digitalcomputer in Fortran IV. The standard fourth order Runga-Kutta method was used for numerical integration of a systemof ordinary differential equations. The computer programwith two terms (Equation 3.11) is listed in Appendix B.The typical output contains not only the profiles of velocitycomponents (u and v), temperature and density but also thedisplacement thickness, momentum thickness, energy dissi-pation thickness, enthalpy thickness, velocity thickness,skin friction coefficient and convective heat transfer co-efficient as a function of the Prandtl number. The Nobel-Abel equation of state was used to account for the imper-fections caused by high-pressure powder gases.
First, an attempt was made to set up tha computer logicand also to obtain tile typical unsteady boundary layercharacteristics with minimum labor. This objective can beachieved by preparation of a computer prcgram with only thefirst term of EQuation 3.11. Moreover, reasonable results
45
were anticipated with just one term. The results discussed
in this section were obtained by this one-term computerprogram. The assumptions involved should be justified beforeinterpretation of the results. The variation of parameterB, for the sample problem in Section 5.1 is shown in Figure 1.This figure indicates that the parameter Bi is not a strongfunction of x, but that it varies significantly with time, t.Therefore, discretizing B1 with respect to x, approximatinglinearly between the nodes (i.e., utilization of Method ofLines), and allowing B18 s to vary continuously with time, t,are justified. Also, B, decreases with increasing x toaccommodate the growth of the boundary layer.
Next, an attempt was made to determine the convergenceof the finite-difference scheme used in the Method of Linesby increasing the number of nodes from 9 to 17. This is thesame as halving the spatial step size, Ax in the computations.This, in turn, doubles the number of ordinary differentialequations to be solved either by standard Runga-Kutta or byothe, predictor-corrector methods. With the same time step(0.25) as in Figure 1 but with halved spatial step size(.0625), the results were unstable. However, good resultswere obtained by halving the time step whenever the spatialstep was halved. Since B1 's and Al's influence any physicalparameter of the boundary layer (and the sample problem inSection 5.1 happened to be an uncompressible one), only B1 'sare listed in Table III.
These results show that B 's change only in the thirdor fourth digit after decimal even though both time andspatial step sizes were halved. If only spatial step sizeis halved by doubling the number of longitudinal stations,the changes in the value of B1 's can be interpreted asnegligible.
The longitudindl velocity component is plotted in Figure2. The present results are shown by solid lines. Hall'sresults are shown by dashed lines. The profiles at the up-stream station (x=l.0) coincided with Hall's profiles fortimes 1.0 and 1.5 even though only one term of Equation 3.11was utilized to obtain the present results. Slightly differ-ent results are obtained at upstream and downstream stations(x=2.0) for larger time (t>7). The difference is believedto be due to the use of different time-dependent boundaryconditions (i.e., matching only the slope at the wall in-stead of the whole profile due to the use of only one termof Equation 3.11) instead of Hall's actual conaitions. Sincethe results are quite satisfactory, even one term of Equation3.11 can bA concluded to yield reasonable results.
46
5 _ _ _ _ __ _ _ _ _ _ _ _ _ _ t1l .0
t=1 .5
4.
B1
.3
t4.
.2
.75 1 1.25 1.5 1.75 2.0x
FIGURE 1 Strearnwise Distribution of Parameter, B1
47
o C ) 0 t C. rf U3 n- C 'J t ) 0 0 I-.
* r-. 0 q~ 01. i M' ,- M C cr-U') U- IC) L ~ C) C)CY) C') C'J Cm C~iCN
IO r-%(7 ot~o L a a Cint-0C
<I r- an q- t C)- P,- () O CM) C ) CI r N
IC) 44 ) 4 l o t o -r 0 tW 10 toN~ I C) M ') nC)M
- O -in Kr qd 'e CV) (Y) CV)N 04 0
LUI-4
9-0 CY)0t a CM LO % CCJ( 4 oc
LC) C) r 0 Lf ' M n )r
LOr- O 0 C roo - 1- 00CY c C-
. . . . ..-
IC)
11 0 ~CV) 0 V)C)r. M ar-Oi)
x * a r a r z m 0)ULo IC) (y~4 ~ ) cy) m' m' (\ C4N
LOC)gCmO oLcan ODto t rl C
LO zC t m c') C) M' en) M) CM \1
x LC) C) in O 0 in0aLn 0uLfEr-. 0 c, a) P. 0D c'.i in r- 0o N-
C- 0% -~j CY N N 'cvl
*.8
.4
-, y0 12 34 5 6 7 8
FIGURE 2 Profiles oil Longiltudinal Velocity Coniponent
49
The transverse velocity component is shown in Figure 3.This is unavailable in Reference 44. However, the profilefor-larger times (t>7) is in agreement with the steady-stateBlasius profile.5'
The streamwise distribution of skin friction coefficientat various times is shown in Figure 4. At time t=l, theskin friction coefficient is uniform over most of the plate.Very little difference exists between the times 4 and 10 up
to x=l.5. Finally, this is in good agreement with thesteady-state Blasius skin frizction coefficients. The tran-sient displacement thickness is shown in Figure 5. The dis-placement thickness increases not only with time, t, butalso with longitudinal coordinate, x. The present resultsare different from Blasius results (dashed) by approximately5 per cent.
The two-term (Equation 3.11) computer program listed inAppendix B is not yet operational. This is yet to be "de-bugged" for reliable results. The numerical results of thesample problem stated in Section 5.2 were not obtained be-cause of the expectation of similar results as above if onlyone term is used. The convergence of the terms in Equation3.11 is yet to be proved, at best numerically, when the two-term computer program is operational. Reasonable resultsare somewhat surprising, with even one term of Equation 3.11.
50
.6 t =7
t=6
.4
.3
v .2
1 2 3 4 5 6
IT
FIGURE 3 Prefiles of Transverse Velocity Component
51
'I
•.45 "
.4 -
.35
.3
t=2
.25 -
.2 __t = 4
.1 .t 1 0
.15 1__1___
.75 1.0 1.25 1 .5 1.75 2.00
x
FIGURE 4 Distribution of Transient Skin Friction Coefficient
52
2.6
2.4 __ _ _ __ _ _ __ _ _
9* 2.2
2.0__ _
1.8
t=2
1.4
1 .2t=1
1.0 0 --
.75 1.0 7.25 1.50 1.75 2.0
x
FIGURE 5 Distribution of Transient Displacement Thickness
53
/
LITERATURE CITED
1. Yalamanchili, Rao V. S., "Unsteady Heat TransferAnalysis for Chosen Ammunition and Gun," ResearchDirectorate, U. S. Army Weapons Command TechnicalReport RE-TR-71-61.
2. Gordon, Sanford and McBride, Bonnie J., "ComputerProgram for Calculation of Complex Chemical Equi-librium Compositions, Rocket Performance, Incidentand Reflected Shocks, and Chapman-Jouguet Detona-tions," NASA SP-273 (1971).
3. Yalamanchili, Rao V. S., "Transient Inviscid Compress-ible Flow Through The Gun Barrel," Transactions ofthe Seventeenth Conference of Army Mathematicians,ARO-D Report (1971).
4. Nordheim, L. W., Soodak, H., and Nordheim, G., "ThermalEffects of Propellant Gases in Erosion Vents and inGuns," National Defense Research Committee, Armorand Ordnance Report A-262 (SRD No. 3447), Division 1,May 1944.
5. Geidt, Warren H., "The Determination of TransientTemperatures and Heat Transfer at a Gas-Metal Inter-face Applied to a 40MM Gun Barrel," Jet Propulsion,April 1965.
6. McAdams, W. H., "Heat Transmission," Third Edition,McGraw Hill Book Company, 1954.
7. Vasallo, F. A., and Adams, D. E., "Caseless AmmunitionHeat Transfer," Vols I and II, Cornell AeronauticalLaboratory, CAL Reports GI-2758-7-1 (1969) andGI-2433-f-I (1967).
8. Eichel Brenner, "Symposium on Unsteady Boundary Layers,"Proceedings of International Union of TheoreticalTnfd Appiied Mechnics, (197!) Laval Press, LavalUniversity, Quebec, Canada.
9. Patel, V. C., and Ni;sb. J. F., "Some Solutions of theUnsteady Two-Dimensional Turbulent Boundary LayerEquations," Proceedinqs of International Union ofTheoretical and Aplied Mechanics, (197l), LavalPreqs, L&'. University, Quebec, Canada.
54
LITERATURE CITED
10. Akamatsu, Teruaki, "On Some Aspects of UnsteadyBoundary Layers Induced by Shock Waves," Proceed-ings of International Union of Theoretical andApplied Mechanics, (1971), Laval Press, LavalUniversity, Quebec, Canada.
11. Woods, W. A., "Some Industrial Problems Associatedwith Unsteady Boundary Layers," Proceedins ofInternational Union of Theoretical and AppliedMechanics, Laval Press, Laval University, Quebec,Canada.
12. Foster, E. L., "A Study of Some Aspects of Frictionaland Gaseous Heat Transfer in Guns," Foster-MillerAssociates, Watertown, Massachusetts, (1957).
13. Anderson, L. W., Bartlett, E. P., Dahm, T. J., andKendall, R. M., "Numerical Solution of the Non-,Steady Boundary Layer Equations with Applicationto Convective Heat Transfer in Guns," AerothermCorporation, Mountain View, California, Report70-22, October 1970.
14. Shelton, Sam V., "Prediction of Wall Heat TransferRates in Non-Steady Compressible Turbulent BoundaryLayers in Tubes," Final Report for Air ForceArmament Laboratory, Eglin Air Force Base, Florida.
15. Dahm, T. J., and Anderson, L. W.o, Propellant Gas Con-vective Heat Transfer in Gun Barrels," AerothermCorporation, Mountain View, California, Report70-18 (1970).
16. Chu, Shih-Chi, and Benzkofer, P. D., "An AnalyticalSolution of the Heat Flow in a Gun Tube," U. S.Army Weapons Command Technical Report RE-TR-70-160(1970).
17. Leech, W. J., and Stiles, G. E., "A Digital ComputerProgram to Determine the Two-Dimensional TemperatureProfile in Gun Tubes," U. S. Army Weapons CommandTechnical Report RE-TR-71-72 (1972).
55
LITERATURE CITED
18. Yalamanchili, R. V. S., and Chu, S. C., "FiniteElement Method Applied to Transient Two-DimensionalHeat Transfer with Convection and Radiation BoundaryConditions," U. S. Army Weapons Command TechnicalReport RE-TR-70-165, June 1970 (AD 709604).
19. Yalamanchili, R. V. S., and Chu, S. C., "Applicationof the Finite Element Method to Heat-Transfer Prob-lems, Part I! - Transient Two-Dimensional HeatTransfer with Convection and Radiation BoundaryConditions," U. S. Army Weapons Command TechnicalReport RE-TR-71-41, June 1971 (AD 726371).
20. Chu, S. C., and Yalamanchili, R. V. S., "Application ofthe Finite Element Method to Heat-Transfer Problems,Part I - Finite Element Method Applied to Heat Con-
Xduction Solids with Nonlinear Boundary Conditions,"U. S. Army Weapons Command Technical Report RE-TR-71-37, June 1971 (AD 726370).
21. Yalamanchili, R. V. S., and Bostwick, S., "NonsteadyFree Convection and Radiation Around Gun Tubes,"Transactions of the Ei hteenth Conference of ArmyMathematicians," ARO-D Report 1972).
22. Dusinberre, G. M., "Heat Transfer Calculations by FiniteDifferences," International Publishing Company,Scranton, Pennsylvania (1961).
23. Lemmon, E. C., and Heaton, H. S., "Accuracy, Stabilityand Oscillation Characteristics of Finite ElementMethod fcr Solving Heat Conduction Equation,"ASME Paper 69-WA/Hif-35.
24. Finlayson, B A,, and Scriven, L E., "The Method ofWeighted Residuals and Its Relation to CertainVariational Principles for the Analysis of TransportProcesses," Chjrical Engi eering Science Vol. 20,1965, pp. 395-404.
25. Finalyson, B. 0., and -riven, L. E , "The Method ofWeighted Residuals - A Review," Applied MechanicsReviews, Vol. 19, No. 19 (1966), pp. 736-748.
56
LITERATURE CITED
26. Kaplan, S., arid Bewick, J. A., Bettis Tech. Rev.WAPD-BT-28 (I1963).
27. Kaplan, S., and Marlowe, S., Transactions of American
Nuclear Society, Vol. 6, 1963, pp. 254.
28. Crandall, S. H., Engineering Analysis, McGraw HillBook Company, Inc., New York (1956).
. "The Method of Galerkin Applied to Boundary-Layer Flow Problems," Developments in Mechanics,Vol. 5, Proceedings of the llth Midwestern MechanicsConjference, 1970.
30. Frazer, R. A., Jones, W. P., and Skan, S. W.,"Approximations to Functions and to the Solutions ofDifferential Equations," Gt. Britain AeronauticalResearch Council, Report and Memorandum 1799 (1937)TReprinted in Gt. Britain Air Ministry Aero. Res.Comm. Tech. Rept. 1 (1937), pp. 517-549.
31. Galerkin, B. G., "Rods and Plates - Series Occuring inSome Problems of Elastic Equilibrium of Rods andPlates," Translation 63-18924, Clearinghouse FederalScientific Technical Information Agency, Maryland.
32. Kravchuk, M. F., "Application of the Method of Momentsto the Solution of Linear Differential and IntegralEquations," Kriev. Sodosbcb. Akad. Nauk. USSR 1,168 (1932).
33. Picone, M., "Sul Metodo delle Miniqie Potenze Ponderatiesul Metodo di Ritz per il Calcolo Approssimato neiProblemi della Fisica-Mathemtica," Rend. Circ. Mat.Palermo 52, 1928, pp. 225-25j.
34. Biezeno, C. B., and Koch, J. J., "Over een NieueneMethode ter Berekening Van Vlokke Platen metToepassing Openkele Voor de Tethniek BelangrijkeBelastingsgevallen," Ing. Grav. 38, 1923, pp.25-36.
35. Ames, 11. F., 'Nonlinear Partial Differential Equationsin Engineering," Academic Press, New York, 1965.
57
LITERATURE CITED
36. Schetz, J. A., "Analytic Approximations of BoundaryLayer Problems," J. Applied Mechanics, Transactionsof ASME, Vol. 88, 1966, pp. 425-428.
37. Snyder, L. J., Spriggs, T. W., and Stewart, W. E.,"Solution of the Equations of Change by Galerkin'sMethod," American Institute of Chemical EngineeringJournal, Vol. 10, No. 4, 1964, pp. 535-540.
38. Kaplan, S., "On the Best Method for Choosing theWeighting Functions in the Method of Weighted Residuals,"WAPD-T-1579 (1963), Clearinghouse, Federal ScientificTechnical Information, Maryland.
39. Hartree, D., and Womersley, J., "A Method for theNumerical or Mechanical Solution of Certain Types ofPartial Differential Equations," Proceedings of theRoyal Society, Series A, Vol. 161, 1937, pp. 353-366.
40. Smith, A. M. 0., and Clutter, D. W., "Solution of theIncompressible Laminar Boundary-Layer Equations,"AIAA Journal, Vol. 1, 1963, pp. 2062-2071.
41. Koob, S. J., and Abbott, D. E., "Investigation of aMethod for the General Analysis of Time DependentTwo-Dimensional Laminar Boundary Layers," Journalof Basic Engineerinq, Trans. of ASME, 1968, pp.563-571.
42. Hicks, J., and Wei, J., "Numerical Solution of Para-bolic Partial Differential Equations with Two-PointBoundary Conditions by Use of the Method of Lines,"Journal of the Association for Computing Machines,Vol. 14, 1967, pp. 549-562.
43. Milne, W. E., and Reynolds. R. R., "Fifth-Order Methodsfor the Numerical Solution of Ordinary DifferentialEquations," Jojrnal of ACM, Vol. 9 (1962).
44. Hall, M. G., "A Numerical Method for Calculating Un-steady Two-Dimknsional Laminar Boundary Layers,"Ingenieur-Archiv. 36, No. 32 (1969).
45. Mirels, H., "BounQary Layer Behind Shock or ThinExpansi : ,ve Moving into Stationary Fluid," NACAN _3 l 2 9 556).
58
LITERATURE CITED
46. Mirels, H., and Hamman, J., "Laminar Boundary LayerBehind Strong Shock Moving With Nonuniform Velocity,"The Physics of Fluids, Vol. 5, No. 1, January 1962.
47. Mirels, H., "Laminar Boundary Layer Behind a StrongShock Moving into Air," NASA TN D-291 (1961).
48. Mirels, H., "The Wall Boundary Layer Behind a MovingShock Wave," in Boundary Layer Research, Proceedingsof International Union of Theoretical and AppliedMechanics, edited by H. GVrtler (Springler-Verlag,Berlin, 1958), pp. 283-293.
49. Mirels, H., "Laminar Boundary Layer Behind Shock Ad- 41vancing into Stationary Fiuid," NACA TN 3401 (1955).
50. Personal Communication with Prof. Mark V. Morkovin,Illinois Institute of Technology, Chicago, Illinois,May 1972.
51. Crocco, L., "Transmission of Heat from a Flat Plate toa Fluid Flowing at High Velocity," NACA TM 690,October 1932.
52. Colburn, A. P., "A Method for Correlating Forced Con-vection Data and a Comparison with Fluid Friction,"Transactions of American Institute of ChemicalEngineers, 1933, pp. 174.
53. Schlichting, H., "Boundary Layer Theory," SixthEdition, McGraw Hill Book Company, New York (1968).
54. Lowe, P. A., "The Method of Galerkin Applied to Flowand Heat Transfer Problems in Unbounded Domains,"Ph.D. Dissertation, Carnegie-Mellon University,Pittsburgh, Pennsylvania (1968).
59
APPENDIX A
Evaluation of Integrals
60
APPENDIX A
I at 4k wfj..(BV) let jv= I e-A2 j dV
e = TV d
ju.civ =Au v-f JV..
--+ A-- -I
B'
2A'"~j
aiFA(Br+A) (A'2)
B1
6)(A13) s5
61
fo g.(A ) e(B,) e- dxt= -
(Ax)= A X - A " A'•x •J0 42 2)6=( x A x + -" - +A T x% -xv + • -6 )C- xl '(3x) x
Tah*n3 each term of +he series separately, o.
obtains integrals AfA XCX Alx(B) d x3 ox e .,.,(Bx) Jx, 7 ~ ),R4(") "x Xe Al x , cx)A1 r" cx A' xe ,, . ,), e1 .
Consider x e v(Bx) =_ Let dv= xae-c x(
Lei- u= ,nLBx), then du.= 2 8e- BXdx
f U V = AkV-f JAI4.fCxeCXiL,(xd:= r a e-
3-p- .j (c x I) eexe dx
62
f -CXtft/BXj _ 3CS ~ 2 8 '3
... xe .i~8XuX -(A. 5)
Consi del- XS_ d X)
Lei Jv= x~ec ~( 2 fa ±pecc
Le + A(~) cuB 8 d
+ x +7- e -4 .)ec+'XJXeCA 4 Bxd c 7 (+~C48
-ac
Let A.'= X' eCdj- -1 9 e~IPecd
xs e c X x +>(z e-cA63
Let AL aqS.(BX) , AA e dx*B
j~xeC~,X6x c -( + + 3 + 3~ e-Ans(B)
+188+ 8Be dA.x
Utu;;z!r n3 6e samne i{ec~nlqu~e as above,
Obe Ohia;ns fx eC A4 (B x) Jx --(2 c)r(c+ gir/ 19+15 c
+ 2 520 C3B 8+ 3024, C' e+ 1712 8GCB + 3848,B] (A.s)
Combining heie !i~eq-a/s, one qe-ts
A ~ ~ ~ ~ A (21C~ 8O&2~ +_C
A3 BS CI:+2 3 84. A- S(IOSC 3+--aOc8+168CB4'++88B6)
A'7 q+(5-C+ +h 2)-;20C" F9a4+324C2 8+ +J1728 C 6 +384. S")
+ ***(A.?9)
46
APPENDIX B
Listing of Computer Program
65
//RAL JOB (----5// EXIC WArF(JR,1I~,E~~~~//WAT.SYSIN UG *$JC'b 'PoB"-iNZK0FLR', IMk=12t ,PA(;'S=16(j,KP=29
IMPLICIT REAL'*8(A-H)qRbAL*8(CI-Z)EXTE~RNAL I-U1,UU2,I-U3pPU4COMM(!N UMEGAl,OMC(,A ,RPL ,ELTA,CPR,DYB,Pl,P2,P3,TWl,TW2CGMMIPN, TW3,TIl,Tl?),Ti ,THT1l,THTl2,TI-T13,RHOO,RH(J1,RHU1l2,kHLl3COMMON~ f.P.H~lOi 1II,;2,rRHIIlj, VTk ,PX,VTHPX,DT19 ,TW2 ,OTW3,VLI ,VL2CUMMUN4 VLj,CUVGA, VVP1, VV,'? ,VVP3,VVPPI ,VVPP2,VVPP3,H,DTI1 ,DT12,DT13COJMMON (ACC'.),,C4'),CC/,Ci-5,CC. ,CCE 3,CCA-5,CC5COMMON 1IPX1,T1PA2,riP XAJ,I)FLTAl,C)ELTA2,JICPCOMMON CrHTl1,1brlil ,-d-rr3DIMENSION iMvEGA(1),AI.ii),A2(12),I31(12),B2(12),AIPCI2),A2P(12)CIMENSLUN hlIPil"),d2P( 1 )T1( 12),TC 12) ,VP(12),XL(12) ,THET1(12)DIMENJSION" ,f~tH12)t,v)P( 12),RHV)(12),D)T(1?) ,DTHT(12),DRHO(12)READ 1, N1, 2,,,4 3, --,4,1N5, N , N 7, NREAD 2, C () =Ih
R --AD 2 , (1P(1), 1=1,2)
READ 2, (XL( I),1=1,Ns)REAO 2, ( T hET 1I 1 tI=I, *6R I 4 L 2, (PRL (1), I= 1,NJ7)
READ 2, (VI4J( I) I1,N"J8)RE4l; 2, ( kCI ( I )vli,~
R cAD) 2, UTH1I I= )I Wh)k E A1) Z2, Ili C I11 (1 1=I, N P)REFAP .3, (C'MF GA I I) 1=I, 11)PRI1N T 1,tN , N t ,14s'),14 0, tJ7, N 8
()R I iT 2, ( I(I1I, 1= 1,41
P R INT 2, (VP(II - I) , H1,PRIr.T 2 t ( XL (I, 1= I1 It41)RI1."j4T 2, ( I W-T I I ) , 1= 1,N.))PRI-N T 2, (PR c( I) I-1=tI I
PRINT 2, (V P P ( I1 1 = IN bPR IMj 2, W(HCT ( I I1, N)
PR INT 2,IM~TM 1, 1=1,Ui1.) R N 1' 2, T H I (1, 1=1 IN
I F OR MA1 (8110 )2 F NM A I F I'~). o)3 EIIRt~iA U.2
GUMGA=. *
SE GA1=I i
[TEST='
Ll=2./LS,dT(PI)(:3=1 ./; I
fC'=tt./.Sf.Rl (III **d/p I
(AE5=-CP/C:Si)RT(P1)
C= 1
fJ 1 1
A2 I )=AI( I), 1P( I)=-.
Ap( I 1
dCIJNTI..Ut:T h 1 =1I.
T'43=1.VTWPX=O..V~hPA=, .1CTW I=- .FTW2=. .L.
T IX 1=,,TI1PX2=.t.T IPX3= C, Cf11 I .bT 12= 1,. 6.r 13= 1 - 6'.V Vp I=5 1 CV VP2=5,,, .VVP3=1)(!VL1=1 C * 7?~VL2=1j.72V L3= 1,. . 71 IT 11= r 1-TWITHT 12=T 1-Ts,2THT13=Tl3-Tl%3P1=2117.P2=2117.P3=2117.V VPP 1= cVVPP2=il..:
67
VVPP3=r.
RhO13"zI I
1'T 12=:.OT 1 i::
rTFI 12= .IJT HTI 1 3=
XS j:: S I T
11 xs i s* 4 xr
CALL L~I~rtR(Ti ,lT llii, ICL LL L I jl.Ai ( TI, 1, Vl, VVP 1 ,1ICALL LI I L ;c (T T T, XL,9 VL I , ICALL LI1'f-:A R( 11 ,TT T 11 II;TlltlCALL L I d:,4 RTT , r, r t,' I)G sL L L F1.C AR [I ?i ViV , VV:'1 t I)CA~LL L I,%E -A{ TII f, % Fit ,RF( fll t I ICALL L I- '.i I (T T I, 1- , I I II - 1)C At.L L ~I(t , r OHil,I.THT I iICALL L I, tA R (f t , F R PH,>H~ 1;!)
3e3 TT=]T+H/2.X S 2:US * F/ + X TXT::XS2(it To 423
422 1I= ICALL L I.tAR (TIr,ii,I 12, 1)(.ALL L I.,[AR(T T , ,V0 VVP2.I)CALL L I~iA R(T T, r.,\L,VL2tlICALL LI"NtAR(1 roi :r, fhTl ,I)CALL L b4EAR ( i T, fP--,P2, I)CALL L I , EAR ( T I, F, Vr 2 , VVif'2 , I)CALL L IN F 4R( fTI fi,.NHlh,.HL'1Z, I)CALL L Il\AR ( IT ,,rbJ,L[ 1 12,1 )CALL L I F 4R ( IT T , 1THT,'PT 141 12 1)CALL L I I-iAR ( 1 f V,f)lif ,rRH( 2,1)
423 T T= T T.I-/2X S 3:: U 5*F-.*+ XTXT::XS3GO 10 523
522 1I= ICALL L I "i EA R( T, T, TI,Ti3, I)CALL LIi4LAR(TI~rTVP,VVP3,T)CALL L~I~AR(TT,I,AL,VL3,I)CALL LI'JIAR( Tl,I, iIICII,THTI3,I ICALL LINLjAR(TT,T, pl CP3,1I)CALL L INAR (rT,I,vP?,vv?Pbil)
68
CirLL L I J 4-( fIIv [,' og-', ',I-'1&5,1CALL LI1. -AR ( I T T,;T,rl 139,1)k.-' L LI: - . I 11T,0TI'T,uTH!Tl ,I
-. 3 Ct.iT I U
I T, I5' j : I T 2 -
CALL .,..(,C,3,, ?i,,2TTllB~,T1,lOtEA,VL1VVP1,TT,1TLST)CA C '4. 1 It)UI1IFST= I(tL. Io 3 *
I F I f)-S 1 (i1
oA2 I )=r,2 I)
35 2 (, t -f 1 141 1
3. 4 CL L K T T h (JI A I ,\ 1QA I PA 2 P b1p FZP)
A = . 125f.
CX2=. ',.I a A t
CeP 1,1,\e
X=X+[,X
F 4T ATUA, 15,4F 1..-5, 2L' ICt
5z5SUM3 I.
,U4(')SS=Cl*' tCA(J M A H J )'r2( J)
CALL Sl-',( Ftll,1, -l?,i- I t 2,rJXl , SLMICELJ=5stMl
GELit= SO",2CALL SIM (Fl3A1 Al, 1,2rXsL3
CALL SIU (I-t,4,A1, ?,& I ,b2,1lX4,SUM4')DLI SUM4PRINi ,DFL39CFLF,,,C,:L2,flGL1
5 CONT INULIF(TT.UT.TTI) 6i fb 110GG TO L'1
1L19 C ONr IN L 7112 CALL EXIT
E NLDSUR~'ULJ I'V- KLI1T (J, X,A'I,42e',IilB2,AlP ,42P ,blP~tb2P)
jiPL ICIT -FLEAHEL&UZ[ 69
C GM -M TW.. ,Tl1 , , 1 1', H lit Fi HF129 rHT13,L.HC' R~H-ll ,R14CI,-,Rr4'1C 1.'V tk tilt., UI I O I 2 -i.'I-II 1- :, VTvPX , VTH~,t (. 1 r ,t TW2 t GT ,VL1 ,V'-2C &'*Mj VL3,:(jG,' G,'Vv,',VVP?,VVP3,VVPPl,VVPIp2 VVP 1,[TlTl,VTl3
COYM'JN' I IP~A1, 1IP 1X, T I.'A ?I L T A 1, 1 FL I j I C~
I Mi' UluN 4f3' 1il 1?) , %K,( IZ ) , A3( 12 ) , AL4"( 12) 1 ,1) ,1( 12) ,A1(?)
G ~I L NIi 'N A - 1 1 4),9A 42 (I.- A : 12,AN4(12)[2 P(1)
A1p(J )=(A1(j )-A!( I-1I) )/1;1 Ia.
hLlPV( iV3V1 ( ; I J - TI 'r1~Il (tNH ... TP
IVL VVP P, Ii IN I1 , C T. Il IPt f1 RaH 1101 1 PTIX C
(JX9bI,"2,i L, ,) (nJlT1'1 ,L ,P I, vv
iC 4*1C *C I i (, J A,9V VL ;' I V 1 4/ )i *4- *-()*2i 2 (J)* )*
F 3=XNUM I/CC'\ l
AK 1 ( J 1-4- FI
ZA2J=21 S RI(L NI ( J 4-)2/ ),4 )(lJ)*'2()*
jElCt*C *' J*~ ( Xf~ (iL 1 , )VV+b/ 1 **) *( t I ( A) *i2+ J J) *'
IC4*CC2*Ujl( J , X,VL , VVP 1 */(ZJ )**2+2 ( J )**2)3r 2:X'Utz1EN' I
AL I U)=H*70
5' AK2(J)=I-4F1-,MU?2 )2C4lU(J , XVL 21VVP2 -2/Z -IIJ**3+Dl CC2 3Ul (JiX VL2 VVPZ)
I;1=C49CC'.*u1(JXVL2,VVPe)**I1/(ZI~(J)*ZB2(J))**3-
F2=x'%LvI*'2f 6A.AL2(J)=1- F2
E TlT2*/DQTA(J)**2e
F3=XF-J. 3/LE-l
1A2(J);*2)*(,%l(J)4* +A (J)**2) )F4=X',JUM4/LEN1AV42(J )=IiF4
1(J)=AICJ)+AM2(J)/2.
1VVPP2, UTI-!12,r4162,i.,T I KHC'1 , I11-X2t[tI;= r I (J, X, Al, At- : I ,,A lP tA2P ~' IP ,32P, P , T~i2 ,TiZ,7Hr!12,
IVL2,VVP;?,VVPP2i,jTII1?,L rW2,1 T12,RHthl2tLRHG1?, 1IPX2)cL;2=FE2(JtXAl,A ,b!,,t.2,AlP,A2Pt ,,B2P,D2,Th2,rl2,THr12,
IVL2, VVP2,VVPD'2,r;TnIl 2,OT6?,('Tl2,RH')1? ,PPHO1 , fIPX4)Xi -Ut/II)I*C5*Ul (J,A,VL2,VVP2:)**2/(ZB2(J)**3)+2Ct*tJ1(JX,VL2,VVP21)*2/(CSC4T(ZL1(J)2+B2(J)**2)*(Z31(J)**2+(32(J)**2))
i,-4=.(*C)tlJAVL,~)-*4(P()Z2J (J,,V2,VP1C*~*S<(JX3LzJt,P2)** )**b1(J )**2J)**3I
F2=XNUM2/CEI.lAL3(J)=H*F2
1A2(J )'h24,AI(J):2+A2J)**%2)F3XNUM 3/LE:4I
XNV=C*[*ITI*)A()*+) 1*CCE3*THT1 **2/U,)S'.2T(AI (J)**2+
F4=XNUMA/UL,'l IAN3( J)=I-*F4
Z(2(J)=v2(j)+4L 1(J)Z41(J )=A I (J ) + Aie3 (J)ZA2(J)=A2(J )+A,*j3(J)C,=FM1cJXl~l.1dE2,ZA1,LA?,01PY,b2PtP3,TW3i,Tl3,THT13,VL3,VVP3,
IV VPP3, 0Th 3, 13, Ki, 1 i,Tl Y3,DRHO 13)
1VVPP3, DTF113, £,T'e3,,, r13,R 13, Tli'X3)C)D1=FC1(JtX,A1,A?,dU,b2,AlPA2PP1P,b2PP.,T63,T13,THlTl3,
lVL3,VVP3,VVP')3,i)Jhr1 3,OTh3',LT13,RHU,13,(AHe13,TLPXj)(CC2=FE2(J,X,AI,A2,tBI,b2,AIP,A2PFIP,B2P,P3,T3,Tij,TTI3,
1VL3,VVPJ,VVPP3,9IIIT13,DTW,[TI3,RH1.,-,L.RHCI3,TlPX3)XNUM1=D*CC5*tjI(J,X,VL3,VVP3)**z/CZB2( J)**3)+D2*(,.t*j1(JXVL3,VVP3
CEN1=C49*C.C5*Lj(J,XVL3,VVP3)**4/(ZB1(J)*Z5i2(J))**3-lC4*CC2*u1(J,XVL3,VVP3)*9.4/(Zbl (J)**2+ZS2(J)*ve)*,3
71
AK4( J)=I-*F 1KNUM2C2*C"49*Ul(J,X,VL3,VVP3')**2/ZBI1(J)**34D1*CC2*U1(JX,VL39VVP3I
1*#'2/(LsUITr(Lr.(J)**2+Lb2i(J)**2)*(ZB1(J)**2+zj,2(J)**2))F' X~IUM2/['L21AL4(J )=F*F2X.NU'13UI.*COt5*THr l3**/A2( J)**3.C,02*CL5*THT1 3**2/ (DSQRT(CAl( J)**2+1A2 (J )-*2 )* (Al(J! 4*+A2 ( J)**2))F3=XN'UM3/[;EN iAM4(J)=H-*F3x %UM4=rCL2*CL3' THT I3**2/A1( J )**3+I)I1*CCE3*THTl 3**2/ DS(,RT (Al J)**Z+
iA2 ( J [A-*2 )'(A HJ )**2+1AZ J )"*e)F4=XNUP4/LEhJl4,44 (J )=H-*F4blCJ)fI(J)+CtAKI(J)+2.*(AK2(J)+AK3(J) )+AK4(J) )/6.32J)=b2CJ)+(AL1(J)+2.*-CAL2(J)+AL3(J) )+AL4CJ) )/6.AICJ)=A1Cj)+(Aml(J)+2.-*(AM2(J)+AM3J))+AJ4CJ) )/6.A~e(J)=A2(J)+(ANI(J)+?.*(AN2(J)+AN3(J))+AN4CJ))/6.
I (1c CU1T INU t:RET U R
40S 0 B RfU T IN EL LI iE 4C(A , X, Y, V V,1IMPL IC IT KEAL*8C A-H1)?Q~FAL*8C 0-Z)CuMMU\' C LGAl,O,EGA2,,PI ,FTA.C,PR,OYBP1,P2,P3,Twl,TW2CLMM("l i~. l1rlI,r13,THiTI1,THTl2,THTI3,RHOcRHL1RHcl2,iHL13C0MM("4 CRHGl,Dr<H[;12,(ORHl3,VTWPX,VTHPXOTW1,DTW2,DTW3,VLlVL2CUMMfI'i VL3,D)OI.A,VVP1,VVI'2,VVP3,VVPP1 ,VVPP2,VVPP3,H,DTl1 ,fT12,CTI3CiJMMO'~ C~,CC51,C4Q~,CC ,CE5,CL3,CCL-3,CCt-5,CC5COMMOGN TlIP X I, IIP X 2,vT P X3, G E LTA I t0E LTA 2,tJ I,C PC UM NON G T HT 11 , D fH1T1 ,;T HT 13fIIE NS I LN XC(1 , Y ( 12)
2 IF (A-X(I)1 3, 1, 1I1=1+1G ToU2
3 1=1-1VV=YCI)*CA-XC 1+1) )/C XCI)-XC 1+1) )+YC I+1)*CA-XCI ))/(X( I+l)-XCI))RETURNENDSUBRUT~INE: 4RCJC1,C.1,A1,A2,O1,b2,VTHT1, VTh,VT1 ,X,VL,VVP,TT, ITEST)IMPLICIT IREAL*h A-It), FAL~f(O-Z)Ci)MMOt. LMLGAJ,OMflGA ,RPI,ETA,C,PR,DYBIP1,P2tP3,TW1JTW2CUMMGN TW3,T1!, Tl2,T1liTHT1II,THT12,THIT13 ,RH0iORHC11 ,RHC12,RHO13COMMON 01Ii1,DRhO1?,P'i[RH(,13,VTPX,VTHPX,DTWI,DTW?,DTh%3,VL1,VL2COMMON VL3,Ct,"(AVVP1I,VVP2,VVP3,VVPPl ,VVPP2,VVPP3,H,nfll,DT12,CT13COMMON C4, CC51,C4s),CC2,CE5,CE 3,CCb 39CCE5,CC5COMMON TIPX1,1 1PX ,TIPX3,PELTA1,DELT 2,Jl,CPCOMMON CTFIT1,ITHT12,( THT13DIMENSICJN AlC 12),A2( 12) ,hlC 12) ,B?( 12) ,XC 12) ,NSIGN(40C )ETA1=.bYY1=.bI11= 1ISIGN=OOEL=.OlIF(TT.GT.1.625) 611 TIJ 636IFI ITitST.EQ.1) GO T,0 E 9
88 b2(J)=.7189 IF(TT.GT*1.) 10 TO 68cl90' BI(J)-1./(2.*OSJrT(Tr)*OME.Al)-82uj*OG2/t'EGAI
F=OMELA1*DERF(C.bO~k*61CJ))+uJMEGA2*jE<FCO.6DG,*B2CJ) )-DFRFC.3Di/IOSORT CTT))PRINT,ti2CJ), F
72
GO 10 12f
121 IF( SIGC,.t-Q.1) GI TCJ 12',
GO 10l 131130 %is162.1 11 )=1I13 1 IFUII-1) 133,133, 132132 1 F V.S G( I I )IS~~ I 1 -1)) 12 9133tl.2.:133 11=11+1I
Cu TCO q-OI~. FP=Cl*YYI1(uME0CA2*C-r.E;XP(-81(J)**2*YY1**2)+UEXP(-BZ (J)**2*YY1**2))
62 (i)=H2(W)-F/FPbl(J)=1./( .*SRT(TT)*(JMEGAI)-ts2(J)*(JMEGA2/OMLGAIISG.N= I
135 PRI!UNT 1369J9,olCJqH'2(J)1.36 FjRAAT(5XqJ3v2I15.3)
IS I('N=
637 1 I.6b9 IIF(TT.C*L.i.c;d.5) '.,0 TO 68F
IF(T.GT.1.i) GO TO 690j
(Al TO 69,.
6WC "I(J)=Cl/tJMPE(GAl*(. 332,,6+(1I./DS0RT(P1*TT)-.332316)*uEXP(-.2s*
F=(i.cGAI*CERFU31l(j)*yyl)+lMtGA2*DERF(82(J)*YY1)-.19394-(DERF(i.3DO/LSJi<1(TT))-.19b,94)*EXP(-25*(TT-1L**2)PRPJTE82(J),FIF(TT.LI-i*5) GO W' 621Gu TO 621:
621 IF(ISIGN.EQ.1) GO. V, C'2IF( F) 6C , 62',63
600 NSIGN(11)=G'GO TO 631
630 NSI.GNC(I I)= i631 1F(I1-1) 633,eo33,6'i2632 1[F (qS IGN I I)-qS I 1;((I 1- 1) 62.,633,t623633 11=11+1
B2 WJ )=b2 (J ) +Ct-LGU TO 69L
62U FPmC 1*YY 1*0OMEGA2*,(-G P (-f 1 (J )**2*YYI **2) +OF XP (--b2 (J) **2 *VY1**Z))b2 (J )=f2( J)-F/FP31,(J)C/UME Al*(.332,6+(1./DS(.RT(PI*TT)-.33206)*bEXP(-.25*
I (TI- 1 )** 2)-C 1*0M'-GA4*A2( J)I'S I GN= 1IF(0A8S(F)-.,"CC-)1) 6359635t690
635 PRINT 136tJBl(J),b2(J)636 CONTINUE
RETURNENDSUbROUTI-NE SIVM(I- AlvA2,81,b2,DX,SUM)IMPLICIT REALt-8(A-H)rREAL*F(O-flEXTERNAL' FCOMMON (0-1EGA 1, 0MvlA2tRvP IETAiCtPRDYB,,PlP2,Pp,Tll tTW2COMMON T63.T 1 1,T 12, 13.ThTl1,THT12,THT13 ,RHOORHOIL ,RHUI2,RHCl3COMMON4 ORHUI1,U RHU)12,PRHU1l3,VThPX,VTHPX,DIWlOrW2,OTW3,VLI,VL.2CUMMON VL3,DCjMGA,VVP 1,VVP2,VVP3,VVPP1,VVPP2, VVVP3 ,ti,DT1I ,0T12.,CT13CUMMON C4,CC51,C49,CC2,CE5,CE3,CCE3,CCE5,CC5
73
CO? I p 1,iT1PX2, T1PX3,L4EL TA1,DELTA2IJ1 cpCO.-fl*'P" GTHTIIDTIITI2t,1HT.3
?U.$~VM4' (,HI, A, A 29F,b l2,VTHT I ,VTW tV TI ,P)
IF(jA0(Tt1PSUM.L-..~..3)GO TO 40'3,10 TLMP=SU1'
StuM=SU+2.r(i-1H)1,At2,Bi,2,VTHiT1,VThVT1,IP)IF(IAhSTLM-$ti).L..(c5)GU TO 4r(,
T LP-P= .UtX=X+ DANp= N P + 1
I F(NP.'*2 * i" il 61.1 5 CA = X+ Ox
J = +
SUMi=SUM+ .*F(,4HujAl,2I31,t ?VTHT1,VTh,VTI,P)5(, X=X+1)'X
S(JM=SUM+F (RHO,1,A, IP1,132, VTHT1,V [k,VTI ti))Xl4p=,NpSUM=X/ (3.*XNP)*SljlPR INT,SUW1
fC, M 0J FU1(GAPHLi1,iA2,,1,TA,VTHT,VTW,VT1,P)t3TWW
C(JMMV1' IVi,T1 9 1, 113,I1jT 119T T12 Th T13,H RHO,, H11 , R12 9 Hr.13COMMO-N U RHO I I, OR HO12,tI'R I ;13,V TWP X ,VTH P X,D T W1, DTW2, D TW 3, V L , VL?2CUMMIUI. VL3,tD~h-'GAVVP1 ,VVP2,VVP3,VVPPI,VVPP2VV'PP3.HOTIllt)TI2,CT13CummO'i C4, C C 51C 4 -3,C U, C I5, CL 3, CC E3 ,CCb 5 CC 5CUMMUN [IPXI,TIPX2, rIPXiIETAItrWLTA2,J1,cPCOMMON~ OTHTII,1)THT12,I;THT13DIME'NSIC\ AI(I2),A2( 12),61( 12),12( 12)FUI=RhO0'/kHf'13*(Dli-R (U1'(J)*X)+l -. F(f2(J)*X)-(DERF(d1 (J)*X)+
2DERF(i32(J)*X) )**3)
FuNCT1i.4 FU2URHrU'.l,,,2,81,-2VTHT1,VTW,Vll,P)IMPL ICIT RIAL~b( A-H), 1LAL*H(O-L)COMMON uMEGA1,OME(;A2,R,PI,E fA,CPR,DYB,P1,P2 ,P3i, ~TWW2COMMON TW3,T11, r12,T13,THT11,THT12,THTI3,RHU,RHO11,RHO12tRHC13COMMON~' LRHC11,DRII'12,I)RH13,VTtX,VTHPX,DTW.d,DTW2,DTh3,VL1 ,V12COM'iLN VL3,C(IJiG4vVVPI,VVP?,VVP3,VVPPI,VVPP2,VVPP3,HQT11,)TI2,0TI3COMMON C4,CC51 9C40,(.C2,tCE5,CE3,CCE 3,CCE5,CC5COMMON T1PxiTIP3X2,TlPd(3,DELTA1,D)ELTA2,J1 ,CPCOMMON ITHfII,DTH1I2,I)THT13
FU2=RHIOLc/iHO13*((fl)F1(i(JX)+ERF(82(j)*X) )*(VTHT1()ERF(AlIJI*XIRETURNEND(FUNrCTIcji~ rU3(K'HlJ,A,A2,I31,,,'. VTHT1,VTW,VTI,P)IMPLICIT REAL*8(A-H),REAL*8(G'-Z)
74
C)Ai lr Y3,T nll A, !.,1,L,,13y,TkT1TH1,H 1 PH3,tRHlt1,r2,H1
C UT.'tJ: - rFI-;I i , P <ht12 tL;R HL 13 , VTVP X , VTH P X vD T1 .D T W2 tUT W3 ,V L1 , VL 2
OM TI. lPXI,T1PX?, TlPX3,rELrA1 ,C-ELTA2,J1,CPA CMMWNo).U LTtI T11 , L, T IiT 12 , f'T HT13
1 1 CI(.', A I i 2)A 2( I) t1( 12) ,82( 12)
1)R F b 2i \J A 2
F ~ Ujc - F U 4(H! 1,A 1?2 , VI t r2 ,V T fI T1, VT W VTI ,PIMP~LICIT ELA-HEL(Z)CCMMLMN o1:vA1,UMe(;AZ,i d,lF A,CPR,DYf3,PIP2,P3,ThlTW2O,MV3i-: Tw3,111,T12,rl3,rHr11,THT12,THr13,RHOO,kHC11,RJlu12,RHL13COMML'i., L RH(II, )Rf I J 1 ,L, H 1t 3V T 1%P X ,VITH VX 9DT 1 , DTW 2,tDT %3 ,VL1 ,VL42CWAK'MCA VL3,IXGA6, VVP, ,VVP2,VVP3,VVPP1,VVPP2,VVPP3,H,DTI ,flTI2,CTI3COMMi. C c. I, C 49, C C 2, C F5,C C 3 , C C E3 ,C C E 7CC 5CLMlNN T1PXI ,T1PX2, T1PX3,DELTA1 ,OELTA2,J1 ,CPCC;MMU\N1; uHT11, E, H1 1 ?, iTH T1I
FU)4=<IC2* cV1HTI/P*(IA Rl(A1(J)*X)+PiR4FtA2( J)*X) )+RHUi *(R*VTHTI/P+
l1UA)-JC(I/R-.Ll13*([,cRF(li(J )*X)+D:-RF (B2(J)*X))
ir [W, ('1 1, Rhf. 1, T iP'X, DrkH(,'l!1PL ILF [ :AL*8( A-Il) ,-AL*8(0l-Z)COMMCI4 1,,[GAIT14 ';Aev,,PI ,[:TAC,,1RO)YB,P1,P2,P3,Thi1,]*Wf
CGMlAir, C (Jii,DAHt)IZ, ,iH'2i3vVT~IPXVTHPX,DTIW1,DTW2,DTW3,VL1,VL2CLUmcl?1 VL3,LTM GA,VVP1 ,VVP?,VVP3,V(VPP1,VVPP2,VVPPJtH,UTli ,(Irl ,cT13
CUMiiC4, CC I,C49),CC2,CI),CC3,CCE 3,OCL5 ,CC5C OM.'UN TPXI,TIPX?, rlPX3,nELTA1,L)kLTA2,Ji,CPCOMMLJN. DlTFTi1,DTH11i2qf,fTHT13
Ci=-.5*r,IRT(2./p)I)*OVJ:GA1*4!2C2=-O 'EC,A**2/P IC3=-lI./[SLJU CT' I)*UM I.uAl*,OU.EuA2C 4=C 3C5=-2 .*uMiFGA i*OM ;A 2/
C?=-1./(PI*OSQRT(PI))*OM;-iGA1**3
C8=-4 .*(,llEGA 1**2*U;MF GA2/( P IxDS(llT ( PI))*C9=-2.*O'.i"GAI**e4(JPI-GA2/(Pl*DSQRT(Pl)) \oG9
C12=-2.*LJME;Ai*fL'MEGA2**2/( Pf*OSJR T( PI 69
C13=2.*(!PEGAl**3/(PI*OS(QRT(Pl))C14=OMEG,61**2*OMEGA2/(Pl*P)SORT(P 1))C15=(l./3.)*C13C16-Cl5C17=-C6CLR=2.*CMU GA1**2*I1MECA2/( PI*D)SQRT( Pl))C19=C9C2V=C8C.21=0C22=-C7CZ3=-OMEGA1**2*llGA/(PI*D)SQRT(PI))C24=OMAEGAI**2*UjMEUA2/(PI*DSCRT(PI))
75
C2~=.*u~A1**0GA 2/( P *PSIg( P-1
C2= .*o ( AP I *DS(RT (P1))
IC 03',=C3C-3 1=4 . 1i4~i u ;lk2*.2/ (P I DSCR T( P1)I
C03 3=- 3 2C34=2.*C3,C .3 5= C9~C36=2 .*uM GA **2-,Iv/(P ISOR T P IC 37=0.33C 38=(,52
C39=-C2
C4I1=MVEG.A 1:CL Ta1/sT( PI)C42IIPLL fA/S(;;'T vPI)
C43=I0f,1:GA i/LSQR f (P1I
C 45=C 4 2/S1-."T (P I
C46=C43C47'=4 .*(j&Fti.(A 1**I 2iA/0 1
FM I1=C1*tUI(JIX VLiVVII')*[iUlJ, X VL VVPVVP)/b(A *r 1(J) )+C2*
F~l=C*L 1J , , L VP *j (J ,Xg L ,VVP )*OY&*B2 ( AJ)/( IJ) *Bj( J) 4122(JV )4-1U +C*JI ( J ,X, VL, VWp) *1(J, X, VLVV) *U PX (M, VL) *DAIAN(
FM13=C8*4,u1(J,XVLtVVP)**2*UIPX(VVP,VL) *(OATAr\(B2(J)/ C1.414*111(J)
181(J )** 2*DSORThi1 J ~ ?J ) **2)))+C9*L ( J X ,VL VV P)**3*j2 P (J
4(CATA+d1(J)/CS(TU31**#(J*12)*2.))/(CUS(*RT(E(J)**2))2)**)
FM14=Cl *u1(J,XtVL,VVP)**3*&1IP(J)'*1DATAN(B2(J)/(1.414*H1(J)))/(
2C11*UI(J,X,VL,VVP)**2*IJIPX(vVP,VL)*(2.*B2(J)/(Bl(J)**2*PI)*UATAN(3P)2 (J DtS '.JRd ( BI( J ~2 +6 2( J *2 )/1,SQ RrUI(J (A* *2 + 62 (JA**2)FM15=Cl2*OlJiJv.XVL,VVP )b2',zP(J)*(ATA(B2(j)/SRT(R1(J)**2.
IB2(J )**2) )/CCSQRfU5(J~()**2+32(J)**2)**3.b2(J)/( kjl(J)**2+B32(J)
3*Bl(J)*(CaTAN(u,.7 1:10 )/(2.'*1.414*bl(J)**3)+l./C6.*bl(J)**3) IFM1L,=C14*u1l(J,X,VL,VVP)*/-*UPX(VVP,L)l31 (J)*(DArA;\j(82(J)f(1.414*1BI(J)))/(1.414*31(J)*'C3)+P2(J)/(Bl(J)**2*(t32(J)**-2+2.*131(J)**22) ))+C15*LJ1(J,XVL,VVP)*4,2*UiPX(VVD,VL)/(t31(J)**2)+C16*U1(J,XtVLVV3P)**3*elP(,J)/(t.,1(J)**3)+C17*Ul(J,X,VL,VVP)**3*1(J)*I2P(J)/4(2.*B1(J)**24H2(J)*,-2)**2FM17=Clb*Ul(J,X,VLVVP)**?*U1PX(VVPVL)1R(J,/(J, (J)*(2.*BI1(J)**21 +B 2 (J)A2)+ C 19* 1( J ,X ,VL ,V VP *3 *h I( J V~2 P ( J U2 ( J 2*2 .2B1(J )**2.B2(J)**z) )+C2f*Ul(JX,VL,VVP)**3*BI (J)*E,2P(J)/( (2.*B1(J).3-*24.82(J)**2)V-C2)+C 1*11(J,X,VL,VvP)**2*LlPX(VVPYVL)/(BI(J)**2)FMle=C22*tj1(J,X,VLVVP)**3*L1P(J)/(Bl(j)**3)+C23*U1( JX,VLVVP)**?1*UlPX(VVP,VL)/(U2(J)"t'1(J))+C24*Ul(J,X,VL,VVP)**3*.42P(J)/(Bl(J)*2B2(J )**2)+C25*U1(J,X,VLVVP)**2*E62(J)*UlPX(VVP,VL)*(DATAN(B1(J)
4/OS(JR1((b(J)**+2(J)**2)/( (0uB(J)**2+b2(J)4**e))**3
FM19=C26*U1(J,X, VLVVIJ)**2*U1PX( VVP,VL)*32( J)*(DAI AN(B2 (J)/DSL.RT(
76
2**2+be-(J)**2)*ii8i(J)**2+2.*t2(J)**2)))+C27*i1(JtXVL,VVP)**3*32(J)3* b113( J)(2 . L 1( J)2?+F2 ( .1*2)*2 )+C 28* L J ,,VL , VVP) **2 *U l?X4V V 1 9V L b 2(J)U3 1 ( J)*2 . 1J **2+ti 2 J)*2)+ C2 9*U I (J ?X 9VLrV V)
1**)II1CIJI(J,X,VL,VVP)**3*(J)*LIP(J)/( (2.*(J)**2+B2(J)**2)**
2)+C32 Ul(J,XVL,,VP)** *u1IPX(VVP,VL)/(BI(J)**2+2.*BZ(J)**2)+C33*
4AUX(JA,VL,VVP)**3*i2(J)*is2P(J)/((1J*2.b()*)*)
FM11tl2=CJ5'U1(J, X,VLVVP)**2*UIJPX( VV0,VL)*b2(J)/(I(J)*(BI(J)**2+
li2(J)**2) )+C36*ui(JX,VL,VVP)**3*B2(J)*blP(J)/(131(J)**2*(Bl(j)**22+B?(J )*'4Z)+C37*1JX,xVLtVVP)**2*UIPX(VVPtVL)/(B1(J)**2+B2(J)**2)3+~~j(tt~VP)**2()(2J* IJ*2B()*))C39*41J IC A, VL V VP )***Y 1(J )FM113=C4Z",Ul(JtXVLVVP)**2*B2(J)*DYB/(Bl(J)**2+s21J)**2)+C41*11I(JX,VL,VVP)*UU(JX,VL,VVPPvVVP)*R*VTHT1*Al(J)/( (R*VTHIIL-TA*)+2c%*VTW)*t3(J***DQT(AI(J)**2+81(J**2))+C42*Ul(J,X,VL,VVP)*'3C-ul(JPX,VLVPtV'P)*R*VTfITl*A2JJ)/( (R*VTHTI+ETA*P+R *vrw)*Bi(J)**2
FMI14=C4:1,U1(J,XVL,VVP)*CU1(JX,VL,VVPP,VVP)*(ETA*P+R*VTW)/((1k*V1HTI+E'A*P.R*VrW)*,1cJ)**~2)+C44*UlcJ,VL#VVP)**2*(lJX(VVP,VLI2**VT14TI: A1(j)/((: *VTHTI+ETA*.P+k*VTW)*t81(J)**2*DSWRT(AI(J)**2+
IA2(J)/(2.*4(42(J)**2+2.*b)I(J)**2 )
1AI(J)/(4.*DSIJ!T(P1)*1I(J)**z2*(A1(J)**2+2.*B1(J)**2)))
F ' )7 =- 2 . * tI (J ) *; * C*IT Av Fm,(; 7
,e*i.VTI , C6 l. *b I(J *3)F M C; 82. *[B31(J )* C2C * IA F MC 8Xl=CAl(J)"*2*( .'41J)**2+6.*B1CJ)**2))/(16.*Bl(J)**/**(DSQRTC2.4
l2il C)(64a1CIJ )*4.*2)* T24il~)*2A(J**))!
X2A()c2+.*2.*A1(J )*oC.*1CJ)**4+4..*A1()(J))V1.-2+.1.A(J)
X4=-AI(J)**,/4.*A(J)*1%.*(.*f31CJ)**2)**+25-.*2.*b1CJ)**2)
1**3*A1CJ )** +3C24.* (2 .*iv CJ )**2) **2*A1(J)**4+1728.*C2.*H1C J)**2)2*AlCj)*46 +384.*A1CJ)'-*P)/( 32.*(2.*B1l(J)**2)**5*)$QRT(AlCJ)**z+
* 32.*tBlCJ)**2)**9))FMSI7=1.I6b (X1.X2+X)+x4)F MG 1= VT P T 1**2*FIM5 11XX1=(AI(J)*A2(J)*C2.*A2(J)*c:2+6.*B1CJ)**2) )/(4.*C .*ilCJ)**2)**2*
I1DSQRT2.*blJ)'**2+A2(J)**2)**3)XX2=-A1J)**3/3.*(A2iJ)*( 15.*(2.*BICJ)**2)**2+20.*(2.*BlCJ)**2)*
lA2(J)**2-18.*A2(J)**4)/( C.'*2.*bl(1J)**2)**3*DSURT(2.*81C(J)**2+A2(J)2**2)**5))
XX32A1()*6.(.*BI(5/.*ACJ* 5*2*2J)**2)4**3+2(J)*C*)/1(J)*2.
2BlCJ)**2)**4*fl'ST(T2.*1(J)**2+A2(J)**2)**7))XX4=-Al('J)**7/42.*CA2J)*C945.*(2.*Ei1J)**2)*'*4+252'.*(2.*B1(J)
1**2)**3, A2(J')**2+1728.*2.*P1CJ)2**A2(J**6+384*A2(J)**8+30 A.*2(2.*Bl(J )**2)**2*A2(J)**4)/C 32.*C2.*til(j)**2)**5*SQRTC2.*361(J)**2+A2(J)**2)**9) )FM51i8=1.16*CXX1+XX2+XX3+XX4)FM518=2.*VTHTl**2*FM518
77
xXXI=(',W *?*2 *A2( J ~ *I( J )**2) /(4. *(?.*Bi J) **2 **2,k2('SQRT(2,*C1(J)**2*A2(J )**2)**3,'
I A2(J ),*2?+8**A2(J )*4 M '* .B~ )*)**SR~ *1(J *+2j
l*(*2*2)**2*tS168.;: b(l HJ)*2+*A 2( J)**2)+8*42(J *7)/(1.(.
XXX4=-(j)**7/42.*(,(J)*(94*(2.*B1(J)**2)**4+2t2,..*C2.*81(J)**
2A(J )**+38.*42?(J ~)/ M32.*( 2.*6 ( J)**2)**5*SRT(2.*bl1( J)**2+3A2(J )**2 )**9)FMv519=1. 16* XX1I+XXX?+XXX3+XXX4)F.:-12=VTHT 1**2", F M519
[tvU3J?*VTI*)+1CJ/(4* a T(I J)/(J)41*2*((J) (2+.*DSCR1(2)*2 I)*
FMi?4=?.*VTI1.Tl*Vfri4)TA ,\(A?(J)/M t.il4*bl1(J) I /(4.*I)S"'RT(2.*PI I)
FMO5=CSVR(2.*P1)*VTIV**2/(16.*B1(J)**3)IFM=-2.*R*C*b I( J )2/?'( F1FML2+F Mi)3+FMC4+FV-,5+FM518flFMOC.1=.5*LSQI'T(2./PI )*AI(J)**2*VTHT11**2/(DSI]RT(A(J)**2+lUI(J)**2)*
I (A11J **+2*2+(J2*i
FM.,3=1./C)S(ki (P[h*Al(J)'*A2(J)*VTHT1**2/(LSuRT(A1(J)**-2+A2c(J)**2+2
FMiOu4=.5'*USQRI(2./PI )*A2(J)**2*VTHTI1**2/([)SQRT(A2(J),;2+bl(J)**2l*I (A2( J)*9:2+2.* I( J )**2))FMJ:-5=1./LS.JZT(IVI)*Vr ,r1*VTh*A1(J)/(AI(J)**2f+2.*fB1(J)**2)
FMC.9=x .*k*C/P* ( M~14Ff., 2+Fk! 4C3+FM f4+F P;5+ FM(,6Ir-M51=FM ;+Ff.)+FM'j6+FM'' 7+F-Y.8FFM1=1./CSJRT(PH)*A(J)*t2*VTHITI**2!(I)SQRT(2.*A(J)**2+b1(J)-%*2+
FFM2=1./USORT(PI)4,A1(J)*A2(J)*VTHTI**2/('SRT(ACJ)**2+A2(J)**'2+ibl(J )**2.L2(J)**2)*(Aj)CJ)**2.i31(J)**2ps2(J) **p))FFM3=1./CS(JRT(PI ),'Ai.(J)*42(J)*VTIuTl**2/(DSURT(Al(J)**2+A2(J)42+
lEil(J )**24132(J)*2*A )*,PlJ*2B()*)
I bZ(J ) *2 *A2J U ~+1J Z*+62(J ) **2FFM5=1./CSQRT(PL)M;AI C)4VTW*VrIir1/(Al(J)**-2+81(J)I-;'44)2(J)**?IFFM6=1./CSQJRT(PI I*A2(J)*VTVTii/(A2(J)**2+31(J)**2+B92(JI)**2)FI=2.*R*C/P* ( FFM 1+FFM2+'FFZ-'3+rFt44+FFM5+FF Wb)
Z1=(AI(J I**2*(2.*A1(J)**?4-3.*(BI(JI**-?+Fs2(J)**2)))/(4.1E31 CJ)*4c2*1B2(J I**2)**2*I:S."T(A1(J )**2+bl(J)*'*2+b2( J)-*2)**3I 4
1 ( 1(J )* +B2 (JV ~ + t. *A I (J) **4I/(8. BI (JA)**2+s2 ( J) e*2 )**3*1S .TC2e.I(J)**2412(J)**2+AI(J)**2)**5))Z3=A1(J)**5/1C'.*(Al(J)*41 1Th.*(B1(J)**2+B32(J)**2)**3+21..*A1 (J)*'*2*
if 8l(J I**2+b2(J I **2 )**2+168*A1(J) **4(fll( J) **2+B2( J)*'*2) I2Al(J)**6)/(I6.*(b(J)**2+R?(J)**?)*4*SliRJ(A1(J)**Z+BlCJ)** +t382(J I**2)**7))Z4=-Al(J)**7/42. (A(J*(45.*(bj*2+b2(J)**2)**t+252j.*A(J)1**2*IIJ)**2+B2(J)t**2)**3+3C.24.*Al(J)**4*(BI(J)**2+b2(J)**Z)**2+21728.*(B1(J)* +r2(J)'*2)**41(J)**6+.38is4.*Al(J)**8)/( i?.*C(.(J)**2+3fb2(J)**2)**5*DSOkT(AI(J)**2+BI(J)**2+P2(J)**2)**9))FM527zj. 16*fZI+Z2t-Z3+24)FFM7-VTHT 1** 2* FM 52 7Zll=,'AI(J)*A2(J)*(2.*:A2(J)**2+3.*(Bi(J)**2+be(J)**2I)))/(4.*(I(J)*
ZZ2=-A1(JI** 3/3 .*(42 (J)*~( 1$. BIJ)**2+Ei2( JI -**2**2+2c".*A()**
78
)h.1(Jl( )* 2 L (J)-2)!-5
I',(?')I(J )** +t2(J )*2)*2+16J. *A 2(A**4*(B1l(JA)**2+62 (J)**2 )+48.*22( J )%b)/( 16.(1( J ),'2+2(A*2)**4 *DSRT( A2 (J)**2+P ( J )**2+
i142-AI((J )**7/4.--2(JJ) -)*)3+3S24!%1(J) **2+et*(f A) +2**2) 25.*2 J
fM528= I . ii*Z 1+LZ 2+.L 34.ZZLFM52FVFFI l*'*2*Fm52 ,
1- 2 ( J )**2 )**29,1SQRT ( A, ( J i*-+B11J)**2+B ( J )*'*2 **3)L Z=-AL (J)** W A(? ( j) 15. *( b IJ )**2+(A)*2 ) **,2+?. *A2 (JA**2*
2A2 W ) *2 +h I (J ) ,b2 (J))2 ) **5))ZL Z3=A2(J )**5/ 1 . *(A2 ( J )*( 1 5. * (P1(J )**2#+2 ( J) **2 ) *93+?1^. *A2 (JA
j*(2* ( *I)'-* 21 iJ )* 2 ) *2-+ 6.*AJ) **4*( R I(J) *4 4t2 (J, -2 )+48~.
3b2 (J )*2 )V**VT(1XFA ( )*21,( J )D$ORT( J) "- *91() f'C.
2 ( Ul ( J **12 (J )**2 ( P(J*:r2M+t I (J ) *2+F 2(A19+Fl-)'))F FMI1=2 .frrHT *VTfWH i1AN 2 J(PI),- I*((I(J)/(LJ1)**2+B1 (J )**2+b (2J.**
FFMI5*z/TC-T1,ekl 6 1 (J*2t(J/)*2 )!'(Fb( J)**2+E2, () ** ) )(.t~
FFM1 5=VT FT(I* DAT(2 41 )*(r'AT ?(A2I( J)/C2rJR( .1(j)2 .*usb(f*2I1*(DbRT ( I( J ) +thC(J)**2))*3 ) +A IJ)/2(.1()*SQR T( P DiJ)1**2*(
32()*2 J)* , ) *(B12J& 2 i?)J)-12) )
FM52=Fl+48+VFM13+F~ 1'FM 11 5=C4t)U IJ 9 tVL 9VV) ***IPX( VVP, VL) *(E T t*Pg*VTW)/ (*VT HT1I+
I1ETA* P+R*VTW ) 1W ) * 2 +C47*FM5 +C4 8111- 52
RETURNENDFUNCT ILG FM2(J ,K, f i ,2 tA I, A2 t8 PL1B2P iP V 1A vVTI , VTHIiT1VL vVVPVVP,
IMPLICIlT REAL'*8(A-H) ,REAL*8(Uj-Z)
CcOMMw4 r ,f11 ,rTi , T13,11F JT 11 ,TH T12 TI T 13 ,RHV , RHO 1 Rl 1(312,1RHC ICOMMON k O ~JL , R THO12 , OR Hb13, VT IP X, VTH P XD10 I CTW? ,Dr W3, VL 1 , VL 2COMMON V L3, DUMGA,9VVP1, VVP 2, VVP 3 ,VVP P 1,VVP P2 9VV PP3,HPD T 11tLTI12,CT 13COMMON C4, CC 51 ,C 449CC i, C F5, C F3 tCC "-3 ,CCE5 ,CC 5COMMON T1PX1,TlPX2, TIPX3,PI-LTA1,PELTA2,Jl,CPCOMMON tTHrIl,DTHTI2,DTHTl3
79
CC I= -fJMIG A I-GA 2/f-SrR HP H
CC2=CC 1CC 3=-2.*GMEGA 1*0M GA21P I
CC4=.5*c~Pr(./p)*tf."GA2**2CC',=-.5*CC4
CC 7=-4 .*(,"EC AI* 1*0V t: A2/ (P I*OSQRTWpH)CCli=.5*CC7C Q 4*'vE U',G2*/ P I *ISQRT IP M
CC li=4.*0MCGA2*2*!/( PI* DSQRT( 2.*P1 MCC 13=-2.*UL6~A2**'3/(I*0 SQRT( PIMCC P=2.;:CrG A!* C cA 2/ ( P13fSQRT (P M
CC1 5= CC 1,CC l16.*1'MbGA I** 2*OM~ A 2/00I1 *SQRT(Pi)CC18.=-CCI7
CC 19=2.- CC 16
CCi? =-CC9pCC22=-CC2lC C2?3=-CC 20
CC2b=-CC24CC26=CC22
CC 28=CC21CC29= 2.*CtMEGA2**./ P v-f'SCRIT( DI1CC3\, =CC2'CC3I=CC2ECC32=CC24CC 33 =CCl 9CC34=4.*OvG42**3/ t-.P *!S.RT( PIlCC35=2.*L,ELA2** 3/ (3 .'-P1*D SQRT(P I)CC36=-CC35CC37=-CC34 iCC38=-OMtGA*(MEUA2*'/(PTI'LSQR T( PI)CC39=-CC3PCC4c.-UM&tA2** 3/ I*CsRT (P1 ICC41=-CCAi.CC4?=2.*LM-EGA1*0fA 1U3A2/P ICC43=UMEGA2**2/P ICC44=CELTA1*'"EA2/S('2l1(0)
a CC45=!ELTA2*(iMEUVA2/OS(,)RT(PI)CC46=UMEGA2/7S-M.TPICC47=CELTAlflt,EGA?/rDS(,RT (P ICC48=CC'i5CC49=CC46CC50=4.*OPEGA1*!1G2IICC51=4.*COMF-GA2'**/P1FMOL=CC1*U1(JX,VLVP)*LU1(J,X,VLiVVPP,VVP) -kl(J)/f32(J)**2*
1DSQRTfB,.(J)**2+B2~(J)**2))IMO3=CC3*u1CJ,X7VL,VVP)**2*1l-(J)*DYHi/(til J)**2+FB2(j)**2)FM04=CC4**j1(JX,VLVVP )*PIL)l(J,X, VL,VVPP,VVP) /(b2(J) **2)Fi5=CC6*u1(J,X,VL,VVP )**2*1)YB/B2( J)Ff.1C6=CC7*U(JX,LVVP)**2*tJPX(VVP,VL)*hl(J)*L)ATAN(I1J)/OSQR1(
IR1(j )**24Ib2(J ) *'2) /Cb2(j )**2*OSt IRT( l(J ) **2+t2( j) **2)
FM07=CC8*U1(J,X VL,VVP)**3*blP(J)*IOATAN(I(J)/DS,,RI (t6I(J)**,e+t2(JIl
80
~-M'CC U I WX,VL VV-)*-?*UlIPX( VVP,VL) *(91CA *DAAK( 2 J)/DS(RT(
~.~CC i'-1( J, Xt I, VVP )**3, CP(J) *MATANdb1 I(JA/DSC.RT(2.*,2 (J)**2I U /(2.--1 .4114X'f2(j)*2)c1(J/ (2. *E2( J)**,2) *U(I(J) **2+2. 32 (J) *2
1L LC)'LI*)/(J tVgV )**3)+3J*(ATi L2(J)/( C,31B(J)**2+2JF-('! ~+~ ( )~ ~) I. I
I=C 12*U J ,X, VL ,V)*2UPX(VVP VL) (DATA N(C. 7C7CBC) (62 (J)i~2)1=C 3*1 ( J, K VL 9VVP ) 4*3*32P (J) PfATAN(C. 7:-7)M) /2.*1 .414*
-~ ~ iI2(J )*4-3 )+1./( u.*B2( J )**-3))FM13= CC 14- 1( Jq, ~VL , VP) *2*UP PA( VVP, VL)('WA TAbJ(61 (j) /DS( FT (L 1(U1**2+B2(J )**2) )/ (SoRT( L,1( J)**2+1,2( J)**2 ) )'-*3)+b I(J) /( BI (J )*,'-+
1!'liCC15 UlJ,XVL,VP)*2 'UIPX(VVP,VL)*b1 (J)*CEiATAt (ii32(J)/LS >RT(
lb I U C)**f1(J )* ) /( ( SRT( F1( J)**2+!B2(JA)**2) )**),1(A( L(J
~-~ CC 17*t(JitX,VL,VvP )*2*U,1P()*LPxV)p/2.L)/1($J) *(S1 (J*
FM1CC228*Uli(J,XVL VVP )-3* 1(J)/(b ( JMB2(J**(W J)**2+J
IFYP CC239*U I(J,A, VLVVP)3 ( J )*i50(JIA B(J2 4 (J)*B2 ()i2I 11.*2 )
FM P'= CC2C*LJI (J, AtVL,VVP )**24u1PJxvVPLA /U1 (J) **2+2(J*z *
1*2 CJ)*4P102 =CCZ7*UI( J, X,VL, VVP))**3*b1(A *LIP(J/X ( JV)*4(2* ( )A' (J)4-2+
FM? 7=CC2*(JJA, VL, VvP)**2'42 ( J)*1P(VVP1,2V()**DATAMBJ )/-2
1/-b? )*21))
FMk22=CC3"U 1 (J , X,VL .VVP) **3v!-2( J) [b1P( J) B1 (JA)**2+2 .*B2 (J ) *e*21**2)
FM2 =CC26*U1(J,XVLVVP)**2*U1tjPX(VVP,VL)*B2(J)/'(h1(J)-(1J.~12.*2(J)* 2))FM31-CC327'U1(J,XVLVV))*43'bl(J)*P(J) /(F2(J)*** -1(J)**2+
1/2.B2 ( )*2()*3+.(.*eJ)3FM24=CC3(*U I( J tXtVL ,VVP )**3;,2 ( J ) *r)1 P(J B ( 1J )*2+2 VI4,2( J-)1**2)iM3=CC3'*U1J, XtVL, VVP )**dip/t2*(J) VP U*B2(A/ l HI)J ,2124C3~1J,(VV)*2UPCvL /(B2 (JJ)**2))FM31=CC32*U1(J,X,VL,VV))**3*B2P(J)/(ls2(J)/**3()***,'1J*
IM32=CC33*Ul(J, XVL,VVP )**34B2(J )*BD A BJ(A)**3)f2(J *
FM33= CC 3'*JI(J,X,VL, VVP )***1P ( B2,WL)(P1)*f(FM34=CC35Li1U(Jt(VLtVVP)**24B1lPJ)(VPV)4*2*2(J) IFi\3CCLC*Ul(JX,VLVVP)*/.*B2PXJ5(VVV) /(*2)*FM4C=CC41*U1(J,XVLVVP)**3i*R2P(J) /(B2(J)4*3)
FM4I=CC42*UI(J,XVL,VVP)**2*81(J)*0YB/(Bl(J)**2+h2(J)**2)
81
FM42=CC43*U1(JXVL, VVP )**2*)Y81f32( J)FM43=CC44,*Ul(JtXVLVVP*I)tJ1(JXVLVVPPVVP)*R*vFiI*A1 (J)/1 ((R*VIHTl+.Ei'AP*VT. *B2( J)**2*rQRT( A( J) *2+,2 (J) ;:2))FM44=CC4*U(JXVLVV)*UI(JAVLVVIPVVP)*R*VTIITI*A2(JI)/I ((R*VThT1I+H IAgP+R*V I v)-2( )**?*SRT( Ae- (JA**2+3 ( J) **2JFM45=CC46*U(JXVLVVP)*U(JXVL,VVPP,VVP)*(ETA*P4R *VTW)/
FM46=CC47-.U1(J,XVLVVP)**C--UIPX(VV 'VL)*k*VThiT1*A1(J)/(62(J)**2
FM47=CC48*U1(JXtILVVP)**2*U1PX(VVPVL)*R*VTI:TI*A2(j)/C((R*VTHTI+I ETA* P+R*VTW )*P2(J) dSOP T(A2(J)**2+P,2(CJ)** ) )IFIP4=CC49*Ul(JXVLVVP)**2*U1PX(VVP,VL) *(LTA*P4-R*VT1A)/(82 (JI**21*(R*VTHT1.ETA*PR*V/TW))F,'-iCI=VTHT 1**2*A H J )2/( (A J ) **2+Bl (JA)**2+-2 (**2 )*DSQRiT(12.1cA(J)**2+bl(J)**2'+F2(J)Y-*))
4 ~ ~ ~ VHI** 2*A2(J )**2/( (AJ )**2+l( J) **2+2 (J) **)*DSHr(12.*A2(J )**2+bls(J)**2+.- (J)**2))FMQOC.3=VJ1TI**2*'Al(J )*t2(J )/( (AI( J )**2+LI( J)**2 +b (J)**2) *S.Rr(IA1(J)**t+A2(J)**2+1l(d))**2+cD2!J)**2))
12.*A2(j )**2+bI(j )**?+ 2(J)*'2))FMSC5=VTHTI*VTWJ*41(J)/(Al(J)**2+B1 (J)**?+h2(J)**2)rm).IQoc=VTTI*1*vwA/-(J)/(A2(J)**+l(J)**2+bd(J)**z)F,1) I=2* * /CSVPEP * M C'1F.'O 2FMC',3FJ-4+Fiv-. J. 5+
Z2=-A1(J)**3/3.*(AI(J)*( 15.(1(J)**2+.2(J)*2)**'+2?O*A1(J)**2*
1(E1(j)**2+i2(J)*2)' *2+168.*A(J)**4*('1(J)**2+32( J)**2)+48.*2 A1(J )**6 /(16b.*( b H J ),*Z+B2( J)**2) **4*'SCRT( AI (CA **2+~1 ( J )**2+3B2(J)**2)**7))
FM517=1. 16*(11+Z2*Z3+',)ZZ1=(A1(J)*A2(J)*(a.'*A2(j)*,*2+3.*(Bl(J)**2+L32(J)**2)))/(4.*(tql(J)*
1Z2=-AI(J)**3/3.*(AZ(J )*( 1'.*(b1(J )**2+B2(J)**2)**2,.2c,.*A2(J)**2*(ILI1I(J )**24U20J)**2)+1 *,-zA2(,1)**4)/( 8.*( R(HJ)**2+B?C(JA**2) **3j*DSGRT I2A2(J)**2481(J)**2+b2?(J)**2)"*5))ZZ3=Al(j)**:)/1O.*(A2(j)*( 1*5.*(BI(J)**24d.2(J)**2)**)+21C,.*A2(J)**2
I*(f31(J)**2+L'2(J )*2)**2+166.*A2(J)**4*(si(J)**2+B2CJ)*~*2)+48.*2A()*)(6*B()*+2J*2*4DQTA()*~lJ*23b2(J)**2)** 7))
ZZ4= -A1I(J )**7/42.*( A2(J )*( 945.*(f I( J)**2+.2 ( j) **2)*)'4+252-' .*A2 (J)1**2*(t31(J)**?+B2(J)** )**3+3r'24.*A2(J)**4*(rf1( j)**2+FP?(J)**2)**2+41728.*A2(J )**b*(b I(J )*2+t,2( J)Y2)+384.*A(A)**P) /(32.*([, (J) *V,2+362(J)**2)**5*D]S<JT(A2(J)*2+1(J)**+F2()**2)**';)FM518=1. 16*( ZZ1+ZZ2+ZZ3+ZZ4)ZZZ1=(A2(J)**2*(2.*A2(J)**2+3.*(bl(J)**2+b2(j)*:) ))/(4.*(B1(j)**Z
1+B2(J)**2)**2*DSORT(A2(J)*4,2+B1(J)**2+B2(J)**2)*.flZZZ2=-A2(J )**3/3.* (A 2(J M 15.*(Hl( J)**2+ri2 ( A **2) **2+20.*A2 (J)**2*
2A2(J )**24i61(j )**2+82(J )**2)**5))ZZZ3=A2(J)45/4.(2(J M 1,L5.*(B H J)**2+h2 (J) 9*2)**3+21L .*A? (J)1*2(1J*e_+2j*2*218*2J*4(IJ*26()")4.2A2(J)**6)/(16.*(BI(J)**2+J2(J)**2)**4'*DS(frT(A2(J)**2+81(J)**2,
82
ZZZ4=-,A?(J)7/42.*( e2( J)*()j45.*(' 1(J) **2+t32 (J)**2)**4+2'52, e*A2(J)
21(2o.*A2(J)**ULI(J)**2+Bi2(J)**2)+3&4.*A2(J)**6)/(32.*(81(J)**2+
./3t2i.R(J)**2+ FT(2(J)**2))*I9I()/(tl(J)**2+t32(J)**2)*(AI(J
FF:'A;l=VTHT14:vrW/DSfKrT()I )*((,ATAN.(A2(.J)/DSORT(B1(j)**2+u2(J)*42))
!/( (I)SQRrCP1(J)**24bi2(J)*e-2))**3)+A2(j)/( CBHJ)**-2+f32(J)**2)*(2'41(J )**2+( 1(J )**2+V,2(J )'*2))
FF2=CSkTUI *(J)*2
FFiv13=CEIMVTT*A(J)/(SRT(P)*(A(J*2+&I1(J)*-":?+82(J)*;-2))
2A I (J ) *-IAi(. * +*2J/C.TP)(2J*-+1J*2B2(J)-*2 ) )
tFM16b=vrHii/(L.*sf, i(pi))*(DATAIIA2(J)/CS(RT(B(J)**2+Hi2(J)**2
rFA2(J )**I~j(J *4+2A(J )*2/(.(A() ) )2()*)*S,.TA(
1b)*h2J2))
FFMi4'=CC41 *1(JXL*" ,VV)t~x/2Ar.()*FM5C182J)*2*DQ,
1A)*2+U(J*J)
FFt1.LuVTHT*VTF)T1*2* (J)**/(2.*2(J)**2+2*2J**)UQT
FFM4 VTHTI*A2(J)*VTh%/(Ae(j)**+2.*82(j)**2)FF1M2vRhr*/(JvTSO (CJ)*))*'FM2+F.i2(J)** )4FF,5FFv
XII(AI (J )**24-(2.9-Al(j)**c-+6.*82(J)**2) )/(16.*L,2(J)**4*[iSORT(2.*
X2=-A1(J )9**j/3.*(A1(J)*(L,.*2(J)*4+4r.*A1(J)*b2(J)**2+b>.*AlWj)2**4)/(b4.*ti2(J)~*'-I (.S(CRT(2.xb2(J)**2+Al(j)**?j)**!))X3=AICJ).*5/1. *(41,J)*(1:5.*(2.*B2J)**2)*3+21).*(2.*b 2(J)**2'
*2)**4*$SRn.2.*Q(J)**2+A(J)**2)**7))X4=-A1(J )**7/42.*(AI(j )*(945.*(2.*B2(J)*2)**4+2520.*(2.*Lt2iJ)**2)
L4*3*AI(J )*-+ '2 .- ( .b ( )* )'2 A ( )* 412 *-2 * 2 J * 22',A(J)*4e3E4.*AI(J)*-*4)/(32.*(2.*B2(J)**2)**5*DSCRr(AI(J)** ?+32.*b2(J )*'*2)**9))FM527= 1. I6*( XI+X2+X3+X4)'1=(Al(J )*42(J)*(2.'*A2(J)**2+6.*F)2(J)**2))/(4.*(2.*e-2(J)**2)**,)'*
1MsukT(2.*CF,2(J ) 4V2+A2(J)**2)**3)
1+d.*A2(J)**4 )/'8.*-( .'*12( J )4:*2)**3*DSQRT(2.*f2( J) 2+A2 (J)*2)*5
63=AIJ )*'*5/lt.*(A2(J)*( lrb.*(2.*B2(J)**2)**3+210.*(2.*82(J)**2*AL-l(J) )**2 18.()**2J*44.A()*)(b*?*2J~22)**4*CSQRT(2.*5,2(J)**2+A2(J)**2)*; 7))k'4=-A1l(J )**7/42.* (A 2( J M ~.2.*L'2( J *42)**4+25eJ. (2. *B2C(J)*
2b-2(J )*2 )2* 2( J)*0/ 2-(Z f2IJ)~4/ -- 2)*5DrT( - 2( *
.- 1=IJ)1., ( *,*.+, -2)) 16. Xlb.*?IJA**4*OSURT(
d2*2 U W2A( 44
P'-4 )=/I-. 6*2 J M ?7+ ~v .0?( ) **'2+IHI)**2*1-15
FdV,=-A2 (J A 2 ( J*/P F,.'I( 1rFMI*P( )FF 11+2 )3+1.(2 L(A*
(J J )I"v-r';2/ESs p*(() **22 J)**.2J)*2 .*2() *FzM14=C 1AVTHTI*4. J ) 4/13tJ1 .)BI )*A J) **+?*D~i(J)I A2I J
.)2 .*TB2 (J )*I*(i2 )*4.)**33 ,5- = .'~~T~~( 16?'-( vok I5+FF-p16+FF- M1
IutFM26+FIM%* 27+ ? SdTIPI t;.i2 *Si e=-M J IF30F FFV I+~ 4I+F,4+1+FF,% I + FFM2+t4+M4+I"'rFM I.= C* FI A*VT I*A H JM3+ ) M CI-Sl,.d )(AI )-22 f-()*
CUM15=Vl-' GA1,I, kT( PI))*( D TAC(A H A /(P1.44P2 (2.3,TdlTW2
1C)*M AO ( ~RO11 (41?,r o3vr( ~THxD~ J0W ,t)**VL ,V'I2 )-22.b( )*
FF M I 7=T1PX,T 1t~2 1PXM Io.T1,IET12,J1,C
C1=-.*C*L*IRFC.JP) FF." 1+'*16[Ml7FmE,2CP/ES(rTCPI3+F )D13+FDEL14p M3=.5C bC21*./OijXt ,VVP 1 J fLLTA5* 2S U -M I P I +FM +Fi -- 1+F 6+FM +FV 2+Fv1SC0 ME2 FM1 .v1 ~l 6+f i7+FVF 9+FV2+FM 2SUV=F2 6+M7P&?c-' (4 ;,4C1FMP/P3+t-4FP5F,36t 3S UW+=CP/ iE:S+HP/ 30+ F 4 +LIA41 +F' 'l , 45 40+FV 48+FVF I2 MI+SU U ;,-
RCTU84
CE I11=C,
C'. 13=Cr@3C 14=-JL?
CI-' lc)=-C2/. *C;I/ IP T~r~ PI H)*OMEA"D T DL TA
Cc2 S~.C/P4 ORT(1I'I) )4.uMEGA2*O1LTA1*.4?Le~i.~~/(pI) kT(P1)*uME A2*DELTA1*DLTA2
Ci-234.,fC/(P'*<)S( RH~' I )FGAI *DELTA I*DEL TA2
C13 =2.*Ci/(PI).)R(,):k~ 1) fT'FG/1*fELrA1**223= XC:I/vP,I)S-I T (P I )* '(a'%A2'*OEL TA 1*DEL TA2
CE233=-.CP/(POl'D(-)'P)*MGA1*D L TA 1*DE LTA2
CC35=-Cd33C -'36= CE28
Ct3d= .5'CL8CC3 I=CE 4Ct4'-CL38CC4ti=CE 19
Ct43=-2.*CP/ ( 1lyb$RT ( PI ))*fJMEGA2*DELTA1 *DEL TA2CE44~=-CE42Ctb45=- C 143CL4b-CE4CE47=2.*CE46CL48=-2 .*CF3*CCE4-)= .'CE2*CCC5v=4 .*GP/ (P 1*O)S(-RT(P I )*I,'EGA1'2*uELTAIC.51=8.*CP/( PI*DSwjRT ('l)* I )~MEGAI*OMEGA2*DELTA1GE52?=4.*CP/(P I*DSrR I ( di ) 4(.-IA2**2*)ELTAICU53=-CE I
CE56=CE53 C
CE57CE54CE5Z=C54/ULL1A2CE 59= C F8/C PCEbC,=-2 .*()MEGA 1*L TA *ELTI. ~(P I*ODSOdtT( PI)C t 6 1=-UMiE G 4 I*C EL I1A 1 / 1,SQ r P1IC E6 2= -2. OM E GA U-L I A1B2Z/P 1 * -R T P ICE63=-2.*OMEGA2*)ELTA1IJELTA2/(PIy,')SQ RT(PI))CE-64=-L)MI,A2DELrA1/DSCRFT(PI)C E65=-Z? .*()ME GA 1~C 7LTA i.-:2/ ( P J. *SUR Tt II1CL66=-2.*fiMEG'*14 -LTAI*)ELTA2/( P1*DS( 'RT( P1))
C[68=CE62CE69=-2.iLME,A2*[)iL TA1*DIELTA?/( P1 *[)$T( P1))C E 7,- -OM EGA 2 * 1,E T A 1 /O S T ( P I
85
CE71= .5GSe1( H2./P)I W'L TA 1**
C :73=CELrTA 1/4tSc1I (P1HCL:74= .5*CSQR f (2.11)1 )9'C.,L f A1**2C F-75= C E.2CE 76=D)L i A UPC f (P I)
F FE =CE,*~VTi-f l*'vz-I)i/ A I(J)FFtE5=CE6*VTI1T 1**A 2(J1 )*)YLb/( A(j ) **2A2 (J) *2 )+C=7*VTHT *DTW/
i(AI(J)**2)FFC-6=C~f*VTI-Tl*V I fPX4,U1(JpXi VLVVP )*(DATA.%( bI( J)/( 1.4l43*A1 HJ)))
FFE-7=CE9*VTPTl*VIHPX*U11(JXVLVVP)*(A2( J)*DAIAN(b1(J)/DSOiRT(IA1I(J )**2+1.? ( J)**2) )i ( 1 (J )*2*DS'?)RT(Al (J )**2+A2(J) *-2) )+631 (J)*
3)IbWJ)**2) ) )FF8CE IV I HT *2*tJ H(JvX,VL VVP )*AIP( J)*(DATM,(PIH (J)/ (1.414*:AI(J)I) )/C2.82b*A1(J)**3)+LI(J)/(2.*A1(J)**2*(B1(J)**2+2.*A1(J)**2))FF:=L' *lH -**; ~ ~ V)* P J) *IflATAN(13i ( J) /OS(R (IAI(J)**2+A2(JI)*'*A,)/((L;S; RT(AI(J)**2+A2(J)**2))**3)4tB1(J)/( I?AI( * 2 ,'( * 2 * I'A )* +l ( J)**2 )))
2AI(J)'**2*CSI"RT(AI(J)*2+tV2(J)**2)))FF~EI = CF13*VFHT1I*VThPY.*U1(JXtVL ,VVP) *(A2( J) *rD4TA'db2 J)fS CRT(I
I AI(J )**2*A2(J )**? ) M Al( J)**2*DSQJRT(A (J) **2+A2 (J) *12 ))+B2 (J)*
2 CT )N(A20 .SRHAI(I))2L(J)9*2 AI )*2[SR A )*
2)))rFE13=CE I,*VTHT***tJ1(j,xVL VVP)*A2P (J)*DA I AlNb2 (J)/DSQRIu(1A1(j)**2+A2(J)**2))f((PSORT(AI(J)**2+i,2(J)**2))**3)sB2(J)/((2A1(J)**2+A2?(J)** )*(A1(J)**2+A2(J)**2+82(J)**2))
FF 1= [ bV H 1t1 ,X LPV)* T;X B )/(l( )* 2 D CR1I AlJ )**2461(J )**2)FFE15=CL 17*VTHT1*1(.), X,VL ,VVP) *VTWPX*32 ( J)/ (A I(J) **2*[)SORT (1A1MJ**2+b2(J)**Z))F FE1b=CEli1VT4T I* 2U113X( VVPVL)*A l ( J) *(DAT 4W(bl (J)/ HI .414* AI (J)1) )/(2.82e4AflJ)**3.+B(J)/(4..*A1(J)**2*Us1(J)**?+2.*A1(J)**2) )FlE17=CE19*VT14Tl**2*bPX( VVP,VL) *A 2(J) *(DATA N( 61 ( J) /DSORT (A1l(J)**2
1+A2(J)**2) )/( (PS.' T(A1(J)**2+A2(J)**2) )**3)+b1l(J)/( (Al (J)**2+2A2(J )**2)*(AI(J )**2+A.( J )**2+$1l(J)**2))FFE18=CE2t*VTiTI**24LjIP#((VVP,VL)*Al(J))*(DATAN\,i2(j)/(1.414*Al (J)1) )/(2.82&*Al(.I)**3)+F2(J)/('4'.A(J)**2*(b2(J)**2+2.*A1(J)**2))FFE19=CE21*VrHT1**2*UIPX(VVPtVL)*A2( J)*([DArAN(B2(J)/I)SQFRT(A1 (j)**2l+A2(J)**2))/( (DS.,R1 (A1(J)**'2+A2(J)**') )**3)+82(J)/( (A1(J)**2+?A2(J )**2)*(A1(J)**2+A 2(J)*'2+iI2(J)**?)))FFE2C=CL2*VTT1**2*PX(VVVVL)*B1P(J)*AI(J)/( (2.*A1(J)*'*2+
lb IIWJ)**2)**2)FFE21=CE23*VrHr1**2*L,1(J,VLPVVP)*1Pf(J)*A2(J)/( (AI(J)**2+
1A,2(J)4*c24tj1(J**,)** 2)FFE22=CE24*VTHTI*2*iPX(VVPVL)*A(J)/(bCJ)*?.*A1(J)**2+Ell(J)**-
12))FF2=E:*~,l**IXVPVL*2J/FIJ*SR(IJ*4IA2(J)**2+iilWJ**2))
FF2=E6VH *2 l J tLVP fl )* ( J) /(bI( J)4*2*(* 12e*AlMJ**2+tlMJ)*2)) )*(
FFE25=Ck27*VTHTI** 'LU1(J,X,VL,VVP)*(UP(J)*A2(J)/(U1(,***
86
F.~=CE~b~1H142;UJ,,VLVV~i.P()*A I(J)/1(12. *A 1(J)*-2+
FFE?=CI:-VT T I ( JXVLVVP) *B1P (J) *A2 C/.( (A I(J) **2+A2(,J)
3--VIITI*2,jI XVtVP)*2' * (J/( 12. 'A 1( J) **24-
F-r 0=CE31*VTf- I l -*2U1( J , X , LtVVP) *i2P( J )*A2 (J)-/( Al(J) **2+
F FI: - C3vrIhT ~VVP , VL) *AI(JA)/( B2 (J)*(2. VAl (J) **2+
* ~ ~ ~ ~ I e-E1C3.,4T12*U IPX( VVP, VL )*4k2( JA I(LN( A) *(A l (J) **2+iA 2(U)**2+1h2( J )-";'rFEz=UvT T I *2T1(HJ,X,VL,VV))*1b2P(J )*Al (JA/(62(J )**2*(
FF3J=(CL---5 *VrI;T 'L**z-kj( J,X, VL VVP) *2P (A)A2 (JA/ (62 (J)**2;(IA I( J )-2+A2 ( J )*,2+'2 ( J )4 4 ?))F1E34=cujo~.vnIT L44-2*1HJ , X,VLVVP) *zj2P ( J)*AI( J) /( (2. *A1 ( J) *2+
132WJ )**2)**2 )FFE35=Ct:. 7*VTHr**W( J,X, VL,Vv*i2P)*A2u )/( (AlI(J) **2+
~F~3,=C ~ I*VHT A( bVVP Vt.)/(A 1(JA)*B1(JA
FFE34=CC4 1v rFT l**21.1 J , XT 'LVVI.) *BIP( J )*A2 (JA/(61 ( J)**2* (jAl (J)**2+A2(J)**2))
FI f1:' ' C &F4 2*V T IIT 1 *2 *IJIP X( V VP- V L ) A Il ( A) -1B2(JAF F F 41 = CL 43* h T *4,2 , X Vt'VL ) *A ( J) / (B2 A *Al (J) **2 +A2 (J1*2)
F 4 3= C It V T 111l* 24- 11J, X, VL, VVP ) *32P (J )*A 2 (JA b2 A * *2*
FF E 44= C L46*V T 14T 1'2*FY1,t/A I( J) Reproduced fromFF 4 = i4 * T~ -',--: h A (J (A )* 2 A (J * 2 best available copy.
rFE4b=C1.46*VI I,r1**2*j<:I-.1-VT1/( PR*RH)**2)FFF47=C.9 *VTT2*J)*e ( J , VT/ (RHJ**2*PR*(Al (J)**2+
FF 48=C :- V1HTPU ( JX,VL VVt)) *2*f(J) **2*VTI *RhO1/
FFE49=CE ,i*VTI-,T1IQ(J, X,VLVVP) ~*1J) *t32 (JA*VTI*Hl/
FFE50=CE52*VT~sT1*tI1(J,X,VLVVP~)*'*2*h2(J)**2*VTI*RI1G1/
FFE1=CE5*VTI f ** 2~F T I*k/ (1*VTIIT1+h IA*P+R*VTW)FFE52=CE54VTHT 1*2*T 1** A ( j) R*VTtT+ETA*P+R*VTV,)*1 Al U)**29*CSQRT ( AM()**2+A,?(J )**2))
FFE)3=CE55*VTIIT 1*I)TI,-( TA*P+R*VTW) /(A I(JA)*2*( R*VTHT1I+ETA*P+R*VTw
1))~TIEAP+*T)
FF5=E7VHIl2llJX, VP*TP**2J/A(J)**21* ;SQ~T (A I(J )'**2+1t2(U ) *'2 )*(f *.VliT1+F TA *P+R* VT ))FFE56=CE!b*VTFT1*kj1H J, X,VL ,VVP) *VT 1PX*(E TA*P+R*VTW)/ (Al ( J)**2*
lik*VT HT 14 TA *P *VTW)FF.7=CE5-VTHiTl**2 ,U1(J,X,VLVVP)*Lj1(J,X,VL,VVPP,VVP)*R*(ATAN(
1131(J )/(1.)4l QRN 4 (J )**/C .1)(J**) MA1 (J)/DS2*(T(AI (J)**A"2P
~3R*VTW))FFE-8=CL6*VTHiTl**2*Lu1(JtX,VL,VVP)*OU1CJXVLtVVPP,VVP)*R*(DATAN(
2**2))/CS%'kT(42(J)**12+C,,1(J)**2))/(A2(J)**2*(Rk*VTHTI+L-TA*P+R*VT%.))
* ~FFE59=CF-61*VT1-TIJ(JXVL,VVP)*UI(JXVL,VVPP,VVP)*i H( J)*(ETA*P+
87
1P*VrW )/(A (J )**2*I)Sf;-( A I( J)**2+F1 I(J)** 2)*(R*VTHF1+ETA*P+R*VTw))FFE6O=CE62- VTHTI**2*U( J, C,VLVVP))*I)U1 ( JXVLVVPPVVP)*R*(DATAN(
ibZ(J)/1.414*41 (JI f/1 .414+02( J)*DAT4*,4(A1l(J)/DSOR I (Al1(J)**2+)2C(J)2*2) )/CS;,KTI(AI( J)**2+[,2( J)**2) ))/(A 1( J)**2*( A *VTHT1+ETA*P+R*V Ih) IFFbt1=Ct.- *VT.T 1**2*ti' ( J, X, VL, VVP) *DU ( JXtVL,VVPPtVVP) *R* I)AT AN(
I1h2(J )/( 1.414*,2(J))/L.il+b( J)*JAAN(AL(J)I)SOR (A2 (J)**2+e2(j)
FFC2=CE6t*VI-.I L1( JX,VP,VVL)FFE62=C-'44-VrH*I( J,;f,VL,VVP) *'.u1( .iXVIL,VVPPVVP) *2 (J)* (ETA* P+
I-- VTW) ( A()**IJI (JH 1)**2+1,2 (J)**2 ) *(,4VTtiT+ErA*P+H*VTW))rFF3=C6vTHTl*2(ls( J,XtVLPVVt)) **2*Uli)X(VVP,VL) *ik*(I)AT AN(b I (J)
2/OSt RTC(A I(J )*92+01U(J) /(A( J)**2*(2VTHT+E A*P+*VTW))
.L/CSW..I(AeJ):2+bl(J)**'))/(M?(J)**2*CR*VTHT1+ETAMP'RVTWI)FFEtb5Cl * VrVTI *t1( J, X, VL, VV)**2*JIPX( VVP9VL) '91 (J) *((ITA* P+
I1*f) AI( -- 2bS:T (J)**21*(P,*VTHTI+ ETA* P+R*VTw))FFL:6(~t6bVTI J 1'* 2-,1! 1( JtXVL , VVP)**2*4UIPX( VVP, VL)*R*(D AT AN(bz (J)
('QQ'TsP(A! (J )**?)+32( J )*2 ))f/(,',I(J) **2*(R*VTIr1i~0LTA*P+R*VlW) )
F; c,l=CE)VTT*2*11( JX,VL,VVP)**2*1PX(VVPVL)*R*(DATAN1(t.2(J)I
FF6=t7.-T, *l(J ,LVP)*2UcX VVP,Vi.) *,(J)*( rrA*P+R*VrW
FFE64=CF71*VTHT 1*"2,11( J ,X, VL,9VVP) *'LUl(JtX,VL,VVPPtVVP)*R/ (A ( J),*g12*iR*VTHI 1+tTA*0+ *Vlh) 3
I (Al(J )**2*LS-T ( ( J )*'+42(J )**2) *?*VTHTI+ETA*P+R,%VTW)FFF,1=CE7?*V*wTI*'I( JXVLVVr))*rUI(JXVfLVVPPVVP)*(ETA*P-R*VTW)
F E72=CL74*V1I;Tl**2":U)I(J,X,VLVVP)**,%*)IPX( VVP,VL)*R/(Ai (J)**,.I*(R*VTH1I+ETA*P+1-Vl))FFE73=Cl75*VTHT14-2;'tJ1( J,X,VL,VV)*'*2*U1DliX(VVPtVL)*42(J)*R/
l(lJ)**~R li),24( *)*KVIT+-APRVWFFEi4=CE70*VTHT1*U1(JX,VL,VVP)**2*UlPX( VV0,VL)*(t7TA*P+R*VrW)/(lA1(J **2*cRP*VrHr1+FrA- p.R*VrW)SUM1=FFI+FF-2+FFL-3+FF - +FF~c+FFF7+FFE-8+FFC9+FF-1u+FFEl 1SUM2=FFE12+F-Ft-13+FF 1'+FFt- 1t+FF,:16+FFE17+FFE18+FFtll9+FFE2C '+FFE21SU3FE2F73F,--+~2 FE2+F2+Fz+F2+F3,tF3SUM4=FFE32+FFL33+FF.34+FFE.35-+FFc36+1FFF374'FF-t.8+FF~-+FF4 +FFE4SU5F[i+ E3FCf+F4+F4+F4+Ff,.+F-9FC0Fi5SUM6=FF[52+FFfE53+FFrC *FF7"5+F56+FFE57+FFE5+FFt59+FFE6 ..+FFF-u1SUM7=FFEb2+FFE63+FFC64E6+FF6+FF1-.FF67+FFE6iFFE69+FFE7,4-FFE71S1JM8=FFt72+FFI73+rFr 1"iE= ( SUP.- +SUto?+Sut3+S1jy +SUt'5+SUM~6+SUJM7f-SU!MP
RE~TURNLN0FUNCTION FE2(J,XA1,A2,il1,E2,A1PA2P,&61I',b2PP,VThVT1 ,VTIII VL,
IVVP,VVPP,CThT1,OTWIJI 1,RHOl,r'PHO1,VTlPX)I MPL IC IT -REAL*8 ( A-Il) 4EAL* 8iW-)COMMON cwEGA1,OME(;A2, .ioll ETA CtPRttrYo~p P P3,Tvl ,TW2COMMON TW3,TI1,TlE,TI3,THT1I,THT12,THT13,RHOUtROl,RO12RH.13COMMON CRIHO11,I)RHO12,r'RIO13VTWPX..VTPX,DrW1,U W2,DTW3,VL1,VL2COMMON VL3;(FOMG4,VVPIVVP2,VVP3,VVPP1,VVPP2iVVPP3,HtDTllTltOT13COMMON C4,CC51,C49,CC ,CE5,CE3,CCE3,CCE5 ,CC5COMMON rIPXlT1PX2, TlPX3,OELTA1,OELTA2,JICPCJP'MON CTHT110rTHTI12,PTHI13I IMtNSION X( 12),A1( 12),A ( 12-),Bl( 12) ,B2( 12) ,AIP(12) ,A2P(12 3DIMENSION r1P(12),B P(12)
88
CC I 1=-C'/ I.S JWl I' )*[ut1. T It1*['ELIA2CC I2=-.I A e. /P I)*(JPlVL IA 2**2C C E j= C C, ICC!-4=-CP/PiCC C- .2 5*CP*I S'., r ( 2. /P I ) "tU-L TA 2* *2
Crjt7=C;,rt4 j~ ) r*(f-LTA2
CCE!=-2.;4(y/ (fi' P-DSwk THI I) )'*L," %I lkI*DE LTA l I'll'ELTA2
CCE~II2.*kCP/P(S' (I ) )*LMfGAl*I)FLTA2*1*2CC 1= ?; P 1 ;-)),TV r.~A *, T D LTACLE13-2.-CP/( Pl*)S P'T(P1I ) l.XE(,A24rf)FLTA2**2CCU-l4=CCI UC IL)-.CC t I~
CCE17---CP/rS(.,T(PI )*jiFGA,'-!*rpLTA;?CCE 18=-CCLe ,
CC E2 e1-CC .12CCel1=-CLUS
CCi.23=2.*~CCL U'
CCF26=CCThC Cl1~7=CCF%)CCL2f&=-CCI 22CC(20=2.*CCF27 1,CC31=-2.*CCE14
CCE 3 2=CC IoeC,I3 3= CCL 21
CCE 34=-CC E2CCCE35=CCL 13CCE 56=-CCL 3kCCE37=-CCL31CCE.,8=CCIE6CCtE39= .5-'CC>7'CC0CC18CCL41=-CCL39CC142=CCE 12CC4=C/ *SjT iG2~)L~y*CCE44=CCE2CCCE45=-CCtA iCC E46=-2 .4CC C'CCL47=-CCB4CCL48=2.*CCt iCCF49=CCL2
CCL5 1=8./ P I *SURT (-') )LM~I OMECA2*DE LTA2*CCCF52=4.*C*LMiGAe'**HLTA2/(Pl*)SORT(Pl))CCE53=-CCL ICCE54=-CCE2CCE5->=CP*LELTA2/D)SekT(PI)CCE56=CCE53CCE57=CCE54
CC E58= CC Ebb
CCE59=CCE8/CPICCE61= CCI16/CP
89
CC E62= CC5'42/CP
CCE6=-,ILG*fL TAP/( St,RF.TUf1I
CC 65=CCE59CCEbb=-;. .vEA *.L T2**2 ( P I*DSQR T(P1)ICCE67=CCE61CC E68= CC F6CCE69=CCt63
CCE71=CELTAI*L'EUIA2/C-SQRT(PJ)CCE72=. )IjSt(,1(2./Pi )*DLLIA2**2CCF73=CCk71/tMELT4!CCE74z-CCF 71CCE15=CCL-12CCE16=C.L 7 1/ L LTA IFFIC-l"VFIITT*IJ/A2J*2PSR(IJ*2A()*)F FE2= CCE 22 VIHI 1*DLTHT 1/( A2( J)**2F FE4=CCE4*V T Fq 1** 2*A 1( J ) *flY)/(A I(J) **2+A2 ( A**2)
1(1.414 *FT*2J *2.1(J )'/AT(A2(J)/OSTH*)(A2(J)*+1(*2))FFE6=CCEtl*VT1,Tl*VTH-PXU(JAVLVVP)*(A1 (J)*DATAN(BI (J)/,SQRT(1Al(J**2+A2(J)**2))/(A2{j)*yr2*DS(:RPT(AI(J)**2+A2(J)**2))+B1 (J)*JCATAN(AI(J)/CSQR(Ae(J**2+L1(J)**2) P/(A2(J)**2*DSC1T(A2(J)**2+:1B ( J )4*2) ))
2(A2(J)**2*9SQRT(A2(J)*-*2+1l(J)**2))) '1**2+A2(J)**2))/(( (,SC1(All(J)**2+A2(J)**2))**3)+31(J)/( (A1(J)**2+2A2(J)**2)*-(Al(j)*'*2,t1(J)**21) PFFE9=CCEII4VTHTI*2*1LI1(JXVLVVP)*A2D(J)*(DATAN(bl(J)/( 1.414*
142(J)) )/(?.2*A2(J)*'*3)+I31(J)/((2.*4A2(J)**?)*(2.*Ae-(J)**2+21 1(J)**2)))FF2' 1O=CCE:12*VTHT'*VTHPX*LJ1(Jt.XVLVVIP)*(A1( J)*DATANCBi2(J)/CS-,Rt(lAl(J)**2+A2(J)*'%z))/(A2"(J)**2*DS(QRT(AI(J)**2+A2(J)**2) )+B2(j)*DATAN(AI(J)/CSC.RI(A (J)**2+.2(J)**2) )/(A2(j)**2*DsQRr*(A2(J)**2+b,2
* 3(J)**2)))FFL11=CC1-13*VTHT1l*VThilx*U1(JXVL,VV))(DATAN(t32 (JP/(l.4i4*A.J)
1))/(1.414i*A2(J)*-*?)+Cl-J)*I-ATAN(A2(j)/OSCRTCA2(J)**z+H2(J)**e1 12/(A2(J)**2*U;sQRr(A2(jI)**2,%2(J)**2)))FFE12=CCE14*VIHTi**2*II1(JXVLVVP)*AlP(J)*(DATAN(62(J)/DSQRT(141(J)**24A2(J)**2) )/(i)Si,PT(A(J)**2+A2{J)**2)**3)+Bzij)/( (Al(j)2**2+A2(J)**2)*(AI(J)**.?+A2(j)**2+B32(j)**2 )FFE13=CCE15*VTHTI**2*1 (JXtVLtVVP)*A2P( J)*(DArAN'(B2(J))/( 1.414*
IA2WJ))/U( 2.82t4*12CJ )4-3)+B2(J )/M( 2 .*42(JA)**2 )*(2. A2 ( J) **22+B2(J)**2)))FFE14=CCE16*VTHTI*Ul(J,X,VL,VVP)*VTWPX*B31(J) /(2.*A2CJ)** *
2 10SQRT(A2(J)**2+Bl(J)**c2))FFEl5=CCtE17*VTHT1*U1(J, XVLVVP)*VThDX*32( JI/(2.*A2(J)**2*1OSQRT(A2(J)**i+rIe(J)**2))fFE16=CCE1ih4VTHTI**2*U1lPX( VVPVL)*Al(J)*(D)ATAN(bl(J)/DSQRT(AI(J)1**2+A2CJ)**2) P/C PStORT(A(J)**2+A2(JI)**2))**3)+B1( J)/( (Al (J)**2+2A2(J)**2)*(AI(J)**2+A2(J)**2+131(J)**2)))FFE17=CCE19*VT~ir1**2*u1PX(VVPtVL)*A2(J)*f)ATA.4CP1 (J)/( 1.414*A2(J)1) )/(2.828*A2(J)**33+131(J)/((2.*A2(J)**2)*(2.*A2(J)**2+31(J)**2)))FFE18=CCE2O*VTHTI**2'UIPX(VVPVL)*A1(J)*(DATAN(B2(J) /rSQRT(AL( J)1**2+A24j )**2))/((USJRI(AI(J)**2+A2(J)**2))**3)+B2(J)/( (AI( J)*22+AI(J)**2)*(Al(j)4 *24A?(J)**2+bl(J)**?)))FFEI9=CCE21*VTHTI**2*UlPX(VVPVL)*A2(J) *(DATAN(B2CJ)/ 41.414*A2(J)
Ifl/(2,828*A2(J)**3)+i2(J)/( (2.*A2(j)**2)*(2.*A2(j)**2+92(J)**2)))FFE2O=CCE22*VTHTI**2*Ul(JXVIVVP)*BIP(J)*AI CJ)/( (AI(J)**2+A2(J)1**2+81(J)**2)**2)
90S
f'E 1 CC3:T1 -2:jI tt~ VP jP )*2(J ('2.*A2 (J)T*2+
FF?=~2tvir*etl (J,XVL tVP) *A i(J)/( B (J)/C Al (J)**2+112( * 2 ! I( * 2 -2 J/(F (J)/(2.A2( j)**2
Ivl(J)*[2(J 4)*2FfE 2 9Z4=CC2*Vri1FI-:(,xtVLVVP); IP( J)*A NJ)/(I(J)
f f-~~~FFF)5CC j27VTiT1**2*uLp I(J VV VL V) * I (J) 13? ( J)/ ( l (J) *2 *( .
11,2 ( J ) * *2 ) r2)
[:F>=C~2V~il"* P(V,,L*lj/B () (A(J)*2+1 A2 (J ) ; *2 + t,4( J ) * *, )ltC2.*A
(J)**2
1A1(J)5*2+A2(J*T )IIT*4L1J XV ,~)*,2 ()A ()(2 *2 J *
FFN~CFP J)/I (J )*(Ai2J)2*(J2(*)**
FF,:17CCLL47*VTHTIX ?'2 Yp./lP (VVJ),L/FlJ*2 )FFE3 6 =CCE48*~I*vru1 'IL) I **3*CV,' 1*IiHO( RH* 2J/(lJ)iAl(J*A(J)** c2c)+A2 (J)4,*2),FFE.-a,)Cu1VTHTI**.UI(JXVLVP)*Ril(J)/
2 *c )*242JF F E'9CC U1* VTHiT )*U1( jl, xVvP) 'L*A*l(J) *12( j)*(I C *+2j)
1* (A2 (J )**2. '*2*82P(VJ )PV)/ :C()*A(JFFE42=CC/4VTHT1Y - I*~(I /(ALVP'2 (J)*A*cvTI1T1 (J*+*vT*,1ICSORT ( A (J )**2i)~2
FFF4lCEJ6*T** 4-2(Y*(R *vrH1,grp+RAl)*,2J*2FFE4 =:C L47':VT~ixt?*IYt / 21+
FF 46 CC 48 VT IT ** * .-, 1J ** *C VT *4 (1 /( II~ j *2 PPI91. T
1IET A*P+R* VTW)FFE56=CCE5"*VTHT1*U1(JXVLVVP)*VT1PX*(ETA*P+R*VTW)/(A2(J)**2*i R* VT HT 1+ET A*P +R*VTW~:))FFE57= (CCE59*VTHTI**2'*U1(J,X, VIVVP1*DUI (J,XVLVVPPVVP)*R/ (R*IVTHTJ+hrA*p+R*vrw) )*(A1(J)*OATAf1(B(J)/DSQRT(Al(J)**2+A2(J)**2) )/2(A2(Jj*-'.2'LVS"RT(AI(J)**a+A2(J)**2))+B1(J)*DATAN(AI(J)/OSQ:RT(A2(J)3**2+I31(J)**2))i(A2(J)**2*DSQRT(A2(J)**2+Bl(J)**2)))FrE58=(CCE6,,*VTHT1**2*(l(JtXtVLVVP)*DUI(JXtVLVVpPVVP)*R/(R*IVIHT1+ETA*P4R*VrW) )*(DATANIf31(J)/.( 1.414*A2(J) ))/(1.4]4*A2(J)**2)+21.1 CJ )'rATAN,(A2(J )/DS.,jT(A2(J)**2+Bl(J)**2) )/(A2(J)**2*DSQRI(3,12(J )** 2 + 3l(J )** z) ))
1VrW)*H1(J)/( (R*VTHiT1+ETA*P+R*VTW)*[OS(RT(A2(J)**2+bl(J)**2)*A2(J)
FFC60=(CCE62*VTtiI1l**?*U(JXVLVVP)*DU1 (4,KVL,VVPPVVP)*R/ (R*
1VTHTI+ETA*P+tR*VTW))*4AI(J)*DATAN(B2(J)/DSQRT(A1(J)**2+A2(J)*4,2))/2(A2(J)**2*DSQRT(A1(J)**2+A2(J)'**2) )+b2(J)*DATAN(Al(J)/DSQ 1RT(3A2UJ) ** 2 +H2UJ) **2 f))IFF-E61=(CCi63*VTHTI**24-i(JX,VL,VVP)*DU1(J,X,VL,VVPP,VVP)*R/(R*1VTHI1+ETA*P+R*Vrw) )*(DATAN (i~2(J)/( 1.414*A2(J)) )/(1.414*A2(J)**2)+2t2CJ)*CATAN(A2(J)/DSoR4T(A2(J)**2+32(J)**2) )/(A2(J)**2*DSQRT(.3tA2(J)**24b2(J)**2 )FFF62=CCE64*VTHri*tJ1(JtXVL,VVP)*PU1c JXVL,VVPPVVP)*(ETA*P+R*VTW
FFE63=(CCE,65*VTHT1**2'*U1(JX,VLVVP)**2*U1PX(VVPVL)*R/(.R'VTHTI+!:TA*P+R*VrW) )*(AI(J)* 'ATAIJ~b(J)/SQRT(Al(J)**2+A?-UJ)**2) )/(A2(J)a2*O*SQRT(A1(J )**2+A2(J)**2H')+B1(J)*rATAC4(J)/SR(A2(J)*Y,+3131(J)**2) )/(A2(J)**2*I)S(,RT(A2(J)**2+131(J)**2)))FFEb4=(CCE66*VTHT1**UICJX,VLVVP)**2*U1PX( VVPVL)*R/(P.*VTH1 1+IETA*P+R*VTW))*(OATANCItlJ)/( 1.414*A2(J)))/(1.414*A2(J)**'2)+B1(J)24*UATAN(A2(J)/CSjIRT(A2(J)**2+Bl(J)**2) )/(A2(J)**2*OSuJRT(A2(J)**2+3i 1 J )**2 )) )rFE65=CCE67*VTHTI*Ul(JX,VLVVP)**2*UIPX(VVP,VL)*(ETA*P+R*VTW)*1F1(J)/(A2(J)**2*O.SiQT(A2(J)**2+Bl(J)**2)*(R*VTHT1+ETA*P+R*VTW))FFE66=(CCE68*VTHTI**2*Ul(JPXVLVVP)**2*UlPX(VVPVL) *R/(R*VT-Tl+IETA*P+R*VTW))*(Al(J)*C,ATAN(b2(J)/DSQRTCAl(j)**2+A2(J)**2))/(A4jJ)z**2*DSQRT(Al(j)**2+A2(J)**2))+B2(J)*DATA%(Al(J)/DSGRT(A2(J)**2+3B2(J)**2) )/(A2(J)**2-KJSORT(A2(J)**2+B2(J)**2) ))FFE67=(CCE6,9*VTIiTl**2*Ul(J,X,VL,VVP)**2*UlPX(VVP,VL)*R/(R*VTHTI+1CrA*P+R*VTW, )*(UATAN(Ib2(J)/( 1.414*A2(J)) )/(1.414*A2(J)**2)+b2(J)*2OATAN(A2(J)/CQRT(A2(J )**2+B2(J)**2) )/(A2(J)**2*DSQRT(3Ae2 (J )**2+b2(J )**2)) )FFE68=CCE7"!*VTHTI*UI(J,XVL,VVP)**2*U1PX(VVPVL) *(ErA*P+R*VTW)*JAH2(J)/(A2(J )**2*)'SQRT(A2(J)**2+82(J)**2)*(R*VTHT1+ETA*P+R*VTW))FFE69=CCE7I*VTHT1**2*U1I(J,XgVLVVP)*LU1(JXVL,VVPP,VVP)*F&*AI (J)/1(A2(J)**2*DSORT(AI(J)**2+A2(J)**2)*(R*VTHT1+ETA*P+R*VTW))FFE7O=CCE72*VTHT1**2*UI(JtX,VL,VVP)*fAJIC(J,X,VL,VVPP,VVP)*R/ (.A2(J)1**2*(R*VTHT1T+ETA*P+R*VTW))FFE71=CCE73*VTHTr1u1(J,X,VLtVVP)*Du1(J,XtVLvVVPP,VVP)*(ETA*P+
iR.*VTW)/ (A2(J)**2*(R*VHT+EA*P+R*VTW))FFE72=CCE74*VTHiT1**2*U1l(J,X,VIVVP)**2*UIPX(VVP,VL)*R*Al(J)/(A CJ)1**2*DSQRT(A1(J)**2+A2 (J)**2)*(R*VTtiTl+ETA*P+R*VTW))FFE73=CCE75*VTHT1**2*tl1(JXVL,VVP)**2*U1PX(VVP,VL)*R/(A2(J)**2*l(R*VTHT1+ETA*P+R*VTW))FFE74=CCE76*VTHTI*U1(J,X,VLVVP)**2*U1PX(VVPVL) *((-1A*P+R*4VTW)/1(A2(J)**2*(a,*VTH11+ETA*P+R*VTW))SUM1=FFEI+FFE2+FFE4+FFEt+FFE6+FFE7+FFE8+FFE9+FFEIO*FFE1 1SUM2=FFE12+FFE13+FFE14+FFF15+FFE16+FFE17+FF.1E!+FFE19+FFE21+FFL21SUM3= FFE22+FFE23+FFiE24+FFE25+FFiE26+FFI27+FFE28+FFE?9+FFE3C.+FFE31SUM4=FE32+FFE33+FE34+FFE35+FF36+FFc37+FFE38+FFI39+FFE4U+FFt41SUM5=FFE42+FFE3+FFE44+FFE5+FE46FFE47+FF48+FF49+FFE5+FF.51
92
SUM6=F FFl;z+F" )j+FF" ;)4+F " 3:+FF u+FFi-57+f F 58+FFE9:+)FF 6-+Ff .I~~~~SuM7: r F -6. +F FI -t- 6., - Fz-.o- +E i-,o:)+F F b6+FFE 07+F F --6',!+F F- y+ FF,_"7..,4--F F;- I I
SU:;$=FF-72+FF-7,+Fr 74FE2=(SUp1+SU.,, +SU ,.+SUM'I+&" "-~-U'+tM+U8. ETURi1 ° 'Jr'c.'qJI
FU4CTIGiN uIP'A(VV, , VL)I tAPL I C II] K[.%L* h, A- " : L 8( - )
CCMMO:N T W , ir 12, r1, T iriIrHT I2 TH f] , 9 ,HO0 1, RHGI 2,RHLI3C(,'OX.Ci,, C.,kI( J I,); ,,i 12,1PR,; } , T 6P XVrNAP X,0TI I.1T W2, I. TW3 , V L1,VL 2
C 0MINI V L3, l ,, VVP 1, VV11, VVP3,VVI)PI,VVPP2,VVPP3,H,nT11, ri2,CT13
CUWM~ C4, ,C IC4 CZ,C5,CC3,CCE3,CCE5,CC5CuMMO 1 1 PX,TIP X2,TI, 3 'E LfA 1(IFLTA2,J1,C
;)
CL&IMON I I- iI, DTIiT I, !Ti 113IPX=,, .
FUNCTION LI(J,XVLVVP)IM4PL IC I[ r EAL48( A-Il) ,~rAL*S3( (-Z
C U0 ,v 1174 0 .tG i11, r i.;, , I fiT , , I R H T1 p T , H 0 ,P R H 01, R H0T2 ,RHL1COM;4.' Tf,J,11IIr?,T1..I lHTI±.,iHTL2,THT3,RHO,RHC111tRHOI2,RHL.I3
C.M L,' ,(); h E,{h011, DHI12, R ,RHf 13, VTWI:)X, VTHPX,DTW1 ,GTW2 ,DT W 3 ,VLI ,VL2
C 0 M ,(1 VL3, C 11G/ ,VVP I, VVi2, VVP3, VVPP1 , VVPP2, VVPP3 ,H ,DT 11 ,{)T 1,0 T 13
COMMON C4, CC 1, C 4, C C , C;5, CF 3 ,CC E 3,CC!-5 ,CC5COMIMN '1PxI,T1 ?X, T1PX3,I'CLTA1,1ELIA2,J1,CPCU.Mv 0:4 DTHT1I,DTHT1,I TIITI3
j GIME.ISIGN X(12)U 1 = I1RETURNENDFUNCTION CUI(JXVLVVPPVVP)IMPLICIT REAL*8(A-H),,EAL*8(O-Z)CUMMON tJMEGA1,UZt-GA2,t,,P ,ETACPRUYB ,P1,P2,P3,TwlTW2COMMUN T %3,I11,T12, r13,THT11,TI112,HTT13 RHOc,RHOiI,RH1o2,RH.13COMMON CRi(JII,DRHO 1,[RHO13,VTPX,VTHPX,DTW1 ,DTW2,DI)'3,VLI ,VL2
COMMON VL3,D(JGA,VVPIVVP2,VVP3,VVPPI,VVPP2,VVPP3,H,DT1,I)Tl2,lOTl.COMMON C4,CC51,C49, CC2,Cb5,E3,CCE3,CCE5 ,CC5
COMMO'q I1 PX1,TIPX2, T 1X3, FELTA1,DELTA2,J1,CPCOMMON ITihr 11, DTIi2,0 THT13DIM :NSIJN X(12) roducede i Y
P ETURN besiENC
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