UNCLASSIFIED

UCVUYY CLMIIC*?IU Of "M P"9 MM Ms

*processing time. An alternate method of the assessment of molecularflux Is presented in which the continuous transition from viscous tocollisionless flow Is approximated by a suitably defined breadkownsurface. The molecular flux incident on a given area of the spacecraft

*surface is determined by Integration of flux from all significant portionsof the breakdown surface. Results are presented for exhaust plumes ofvarious stagnatlon and exit plane conditions emanating from anaxisymmetric ring nozzle.

S, MN OI02-LF 0 14- 6601

2 UNCLASSIFIEDS9CumITV CLAMICATION OF THIS PAGIMM" 00 ttoe a..

a, * * *.

Approved for public release; distribution is unlimited.

Molecular Back Flowfrom the Exhaust Plum.of a Space-based Laser

by

Scott E. McCartyLieutenant Commander, United States Navy

A.B., Stanford University, 1976

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLJune 1985

Author: __ _ __ _ _ __ _ _ __ _ __ _ _ __ _ _ __ _ _

Scott E. Mcdarty

Approved by:A.E. Fuhs, Thesis Advisor

M.F. Platzer, Chairman,Department of Aeronautics

: / j.N. Dyer,

Dean of Science and Engineering

3

Q.5

S/ ABSTRACT

Back flow from the exhaust of a chemical laser in low earth orbitmay be detrimental to the integrity and operation of the selfsamesystem. Difficulties arise in the calculation of exhaust plume propertiesand molecular flux as the gas expands from continuum to free-molecular

flow. The solution of the governing Boltzmann equation is exceedinglycomplex; similarly, numerical solutions such as the Direct SimulationMonte Ca , te~pniue requirg prohibitive amounts of computer

processing --- 7 -. alternate method of the assessment of molecularflux in which the continuous transition from viscous tocollisionless flow is approximated by a suitably defined breakdownsurface. The molecular flux incident on a given area of the spacecraftsurface is determined by integration of flux from all significant portions

* of the breakdown surface. Results are presented for exhaust plumes of

various stagnation and exit plane conditions emanati g from anaxisymmetric ring nozzle. )r ,f,(/" ' ,,)

4

%°

% :% .!,*

TABLE OF CONTENTS

I. INTRODUCTION. ....... .................

11I. METHODOLOGY. ........... ........... 11

Ill. SOLUTION .. ......... ................ 15

A. EXHAUST GAS PROPERTIES. ... ........... 15

B. PRANDTL-MEYER EXPANSION .. ............. 15

C. THERMALLY SCATTERED FLUX .. ........... 17

D. DETERMINATION OF BREAKDOWN SURFACE .. ..... 19

E. BACK FLOW GEOMETRY. ...... .......... 22

F. SPATIAL INTEGRATION OF FLUX .. .......... 26

IV. RESULTS. ........... ............... 29

V. CONCLUSIONS .. ......... ............. 37

VI. RECOMMENDATIONS. ...... ............... 39

APPENDIX A: ON THE GRADIENT IN CENTERED

RAREFACTION FAN .. ....... ......... 40

APPENDIX B: EVALUATION OF INTEGRALS .. ............ 42

APPENDIX C: FORTRAN PROGRAM 'FLUX'. .. .. .......... 44

LIST OF REFERENCES .. .............. ........ 52

INITIAL DISTRIBUTION LIST ...... ............... 54

5

LIST OF FIGURES

1.1 Spacecraft Laser ........ ....................... 8

2.1 Prandtl-Meyer Expansion ...... ................. .112.2 Exhaust Flow Breakdown ....... ................... 12

2.3 Contours of Constant-P ..... ................... .13

. 2.4 Extent of Significant Portion of Breakdown Surface as

Defined by the Limiting Mach Number ..... ............ 14

3.1 Flux Geometry ......... ........................ 17

3.2 Flux Ratio versus Back Flow Angle, # . . . . . . . . . . . . 20

3.3 Back Flow Geometry. .................... 23

3.4 Back Flow Geometry: x-y plane ................... .. 23

3.5 Back Flow Geometry: y-z plane .... ............... .244.1 Molecular Flux in Plane of Nozzle Exit, M = 3.0 ........... 32

4.2 Molecular Flux in Plane of Nozzle Exit, M = 4.0 ........... 33

4.3 Molecular Flux in Plane of Nozzle Exit, M = 5.0 ........... 34

4.4 Sensitivity of Flux to Breakdown Parameter, P .......... .354.5 Flux from Actual and Approximated Breakdown Surfaces . . 36

5.1 Exhaust Nozzle Modification ...... .................. 37

5.2 Exhaust Nozzle Modification ..... .................. 38

.oO

-6

ACKNOWLEDGEMENTS

I would like to thank my thesis advisor, Dr. Allen E. Fuhs, for

his insightful guidance and assistance from inception to completion of

this work. Additionally, I appreciate the timely suggestions and

comments of Dr. Joseph Falcovitz during the latter stages of this

project. Most of all, I wish to say thank you to my wife JillAnne,

whose unflagging corfidence and love are ever my source of inspiration.

My love to you, Jill.

-p

7

9,1Z.. .. .. . .

I. INTRODUCTION

Advancing satellite and spacecraft technology will soon enable thedeployment of an array of space-based laser systems. Among the lasers

being considered are chemical lasers of the hydrogen halide class.

Before their deployment may be realized, however, important questionsremain to be answered. Major concerns are the possible contamination

and system degradation of low earth orbit spacecraft by laser exhaust

back flow. One design of a spacecraft laser is that of a long cylinder,its power output device located at one end, as in Figure 1.1.

Ambient atmosphere

Exhaust gas Disturbance to laserWall

contamination'.2 ... .. .... ...

Laser beam

Ring nozzle

Figure 1.1 Spacecraft Laser.

The exhaust gas is released through a ring nozzle and undergoesrapid three-dimensional axisymmetrical expansion. However, a number

of molecules in the expanding plume may have thermal velocity compo-nents directed back toward the spacecraft of sufficient magnitude toovercome the mean flow velocity. As certain molecular species in theexhaust plume of a chemical laser may be corrosive to the spacecraftsurface, the molecular backscatter from the jet must be determined toassess potential hazards to the spacecraft.

8

..................

Considerable difficulty arises in the analysis of the flow field of

the exhaust jet as it expands rapidly from continuum flow conditions at

the nozzle exit plane into the near-vacuum conditions of low earth

orbit. In the region where an exhaust plume transitions from

continuum to free-molecular flow, a microscopic approach must be taken

in the flow analysis. Properly, temporal changes in molecular distribu-

tion functions in physical and velocity space must be related to molec-

ular collision parameters. In such analysis the only applicable equation

is the Boltzmann equation [Ref. 1], viz.,

" )+uO +FX -(nf) = o f n(f'f',-ff1 )g dfdu (1.1)at 0

Here, n is the number density, f(u 1 ,x 1 ,t) is the normalized distribution

function, ui is molecular velocity, a is a differential collision cross

section, dQ is an element of solid angle, FI is an external force per

unit mass, and du is an element of velocity space. The subscript 1

denotes the molecular collision partner and the prime denotes a post-

collision value. The left-hand side of the Boltzmann equation repre-

sents the change in the number of particles in a phase space element

from convection of molecules across the surface of this space due to

molecular velocity ul and external forces F, . The right-hand side of

the equation represents the change in the number of molecules in a

phase space element due to particle-to-particle collisions. The

Boltzmann equation has been the subject of extensive investigation;however, the complexity of the problem has thus far yielded analytical

solutions only to simplified models of a specific nature [Refs. 2,3,4].For instance, the Chapman-Enskog method of solution of the

Boltzmann equation assumes the distribution function f is perturbed bya small amount from the Maxwellian form. The distribution function is

then expressed as a power series in e, a parameter regarded as ameasure of mean collision time [Ref. 51.

Approximations to the Boltzmann equation by Hilbert, Burnett, and

others consists of expanding f in powers of ka, where X is the meanfree path at a point in the flow field and a is a typical macroscopic

9

* * . -.'. .. ,- - , -. • ", -" , • .o .=..- . 2

I7

dimension of the flow geometry. In the first approximation to the

Boltzmann equation, the left-hand side of equation 1.1 is omitted, and

the result is a nonlinear integral equation. Higher approximations

satisfy inhomogeneous integral equations [Ref. 6]. However, the

validity of the above methods is confined to flow of small Knudsen

numbers, i.e., a>>X.

A numerical approach to the solution of the Boltzmann equationproposed by Bird [Ref. 7] is the Direct Simulation Monte Carlo (DSMC)

method, which entails the computer modeling of a real gas flow byseveral thousand simulated molecules. Continuum flow solutions (e.g.,

the method-of-characteristics) are used to compute flow field properties

to the point at which continuum flow theory is no longer valid.

Beyond, flow is simulated on a microscopic scale, molecule-by-molecule.

Velocity components and spatial coordinates of each of thousands ofmolecules are modified over small increments of time until a steady-state

solution is determined. A shortcoming of the Direct Simulation Monte

Carlo technique is that such a solution may be obtainable only after agreat number of iterations of molecular collision algorithms, requiring

prodigious amounts of computer processing time.

In order to significantly reduce the computational time required for

molecular back flow analysis, while obtaining results of the correct

order of magnitude, an alternate approach to the problem is presented

herein.

10

.

a . . . . . . . . . . ..,

II. METHODOLOGY

Consider the gas jet of Figure 2.1, exiting an axisymmetric ringnozzle into a near-vacuum. The gas undergoes Prandtl-Meyer expan-sion in which the flow is turned, Mach number increases, and densitydecreases until ambient pressure is reached.

Yf \

Figure 2.1 Prandtl-Meyer Expansion.Region of Figure I is expanded in this Figure.

At some point along any streamline in such a flow, there is a break-down of continuum flow theory (Ref. 8]. Although the transition fromviscous to free-molecular (collisionless) flow is actually continuous, aboundary where the transition begins may be approximated in threedimensions by a breakdown surface. We wish to evaluate the magnitudeof molecular efflux from the breakdown surface toward the surface ofthe spacecraft. In reality, molecules crossing the breakdown surface inthe direction of the spacecraft enter a transition region in which molec-ular collisions are highly likely to occur, as depicted in Figure 2.2.

J.

11

Continuum Transitionflow region

:Exhausts lS Free molecular

S..(collisionless) flow

.............................................

Figure 2.2 Exhaust Flow Breakdown.

However, within the transition region, translational equilibrium of the

expanding flow ceases to exist, and thermal properties of the gas, in

effect, remain as they were at the onset of breakdown. Therefore,

although molecular collisions occur between the breakdown surface and

the free-molecular boundary, it is assumed that effusion of molecules

toward the spacecraft from the breakdown surface is equivalent to that

of the transition region. Hence, this analysis is concerned with deter-

mination of flux incident on the spacecraft from a defined breakdown

surface.

Bird [Ref. 9] has defined a "breakdown parameter" for steady flow

as

P -s" (2.1)Pv ds

where v is collision frequency per molecule, p is density, u is flow

speed, and s is distance along a streamline. The breakdown parameter

is thus a modified Knudsen number relating mean free path to density

scale length. The onset of breakdown occurs when the value of P is

approximately 0.05. We define r as the distance from the fan vertex in

12

- s_ -. .07 - .. .

the Prandtl-Meyer expansion and show that P may be expressed explic-

itly as a function of r and Mach number. The locus of the breakdown

surface may thus be determined. The nozzle lip, at which point the

Prandtl-Meyer expansion waves are centered, is a singularity point

which is also the origin of lines of constant-P, as shown in Figure 2.3.

P.0"- P 5

Figure 2.3 Contours of Constant-P.

As the nozzle lip is approached, uncertainties in flow field analysis

increase. Here, we are concerned only with flow properties of the gas

located more than a few tens of mean free paths from the nozzle lipsingularity. As the gas flow proceeds through the Prandtl-Meyer

expansion, the value of r on the breakdown surface increases as thelocal Mach number increases. Properties of the flow at any point along

the breakdown surface may be determined as a function of stream flow

speed. Further, the molecular flux from a given element of area on thebreakdown surface is a function only of local Mach number and the

angle between the flux vector and mean flow direction. Thus, from any

element on the breakdown surface, the molecular flux incident on an

element of area on the spacecraft surface may be determined.

At some point along the breakdown surface, the Mach number is

"" sufficiently high that the backward flux effusion toward the spacecraft

13

surface is virtually nil, and consideration of flux from points further

out on the breakdown surface is unnecessary. This point is shown in

Figure 2.4 at the intersection of the breakdown surface defined by

P = 0.05 and the expansion wave corresponding to this limiting Mach

number.

P = .05 //

'U[:: ~............................................-,,- x

Figure 2.4 Extent of Significant Portion of BreakdownSurface as Defined by the Limiting Mach Number.

The total incident flux at any point on the spacecraft surface is calcu-lated by integration over all those portions of the breakdown surface

from which significant flux is received.

14

.....................................

Ill. SOLUTION

A. EXHAUST GAS PROPERTIES

In the schematic representation of a low earth orbit spacecraftlaser system of Figure 1.1, pertinent dimensions shown are the space-craft ridius R1 and half-width of the exhaust nozzle X1. The exhaustjet of a hydrogen fluoride laser consists primarily of four molecularspecies: hydrogen fluoride, hydrogen, deuterium fluoride, and helium.Laser cavity molar flow rates and gas properties [Refs. 10,11,12,13]assumed for the calculations which follow are weighted averages repre-sentative of typical laser cavity parameters rather than those of aspecific operating point. These parameters are as follows:

molecular weight, .4. .. .......... 7.4 kg/kmolemolecular diameter, 6 .... ........ 2.52 x 10-18 mspecific heat ratio, I .... .............. .. 1.493stagnation density, Pt ......... 0.01 to 0.1 kg/M

stagnation temperature, Tt ............. .. 2200 Kexit Mach number, M. ............. ... 3.0 to 5.0

B. PRANOTL-MEYER EXPANSION

Laser exhaust gas exits the spacecraft from a centrally-locatedaxisymmetric ring nozzle and undergoes two-dimensional isentropicexpansion. The following relations are assumed valid at any point inthe isentropic flow field [Ref. 141:

p = pRT (3.1)

R = J t/,(3.2)

A1

". 15

.4 ~V~'~.~~ ~~ -... +i . . ..

[4.

Tt = T[+(y-)M 2/21 (3.3)

Pt = (3.4)

a =-y (3.5

Local temperature, density, and Mach number are denoted by T, p, andM. The stagnation temperature T, and specific heat ratio T are eachconstant in the flow field. R is the gas constant per unit mass, A isthe universal gas constant, and a is local sonic speed.

Figure 2.1 depicts a streamline of exhaust flow, with Mach numberMg at the nozzle exit plane, undergoing Prandtl-Meyer expansion as itleaves the nozzle. The angle between the streamline and a given Machwave is a function of the local Mach number, related by

sin j, = 1/M (3.6)

Also,

•-0 + , (3 .7)

In equation 3.7, 8 is the angle between the spacecraft surface and theflow streamline; 0 is the angle between the spacecraft surface and thetail of the Prandtl-Meyer expansion fan. Owczarek [Ref. 151 gives thetotal angle that the flow is turned from its direction at the exit planeas

0,-02 = -(v,- 2 ) (3.8)

where the angle of turn is equal to 0- 02. Values for v are given bythe Prandtl-Meyer function

.'(M) = _ an-V/(M_1) )/(7 +I) _ tan-V/ T (3.9)

In Figure 2.1, O6= 90 and M,= Me.

16

,•o

C. THERMALLY SCATTERED FLUX

It is assumed that the molecules in any small element of volume inthe expanding flow possess a Maxwellian velocity distribution. Noeller[Ref. 161 shows that the fraction of molecules in such a volume elementthat fly into the space angle dw, as depicted in Figure 3.1, is

(dn,./n r)d 3 ,dc = iT-3/2c2e-(CZ+U2-2cu cos i) dcjdc (3.10)

Y U

FT ~dO

p dc

.....................................-.- x

Figure 3.1 Flux Geometry.

Here, u is the stream velocity, c is the inertial flow velocity (i.e.,thermal velocity plus stream velocity), and # is the angle between the

direction of flight and the mean flow direction. Flow speeds in thisanalysis have been non-dimensionalized with respect to the most prob-

able molecular thermal speed, cm, where

Cm = 2T (3.11)

Thus,

u = Ma3/v = Mv 7/T/2 = f-/M (3.12)

17

a .. . - .

The number density of molecules reaching a point P from the volume r

is

dnp = (dn,/d.)dfl (3.13)

where d2. is the solid angle of the surface F from P. Analogously, wemay define the incident molecular flux in the vicinity of P arriving fromthe surface F as

dip = (diF/do.)dfo (3.14)

* - where

(d!F/dcJ) = l.7Cm7T3/2C3e-(C2 +U2-2cu COS '*)dc (3.15)

Hence,'

(dip/dol) = .i.Cm T3/2C3e-(c2+u 2 -2cu CO *')dc (3.16)

The evaluation of this integral is given in Appendix A. Thus,

dj~= lTm-3/2 u[ (I+U2COS2 *)/2+ (V,7t/2)(u cos

(e U2C02 #')erfc(-u cos .)(U 2COS 2 ~'+ 3/2)] do (3.17)

The local number density is given by

n = p (N/A) (3.18)

where N is Avogadro's number. From equations 3.4 and 3.18,

pt (N/ 4)

18

!0

From equations 3.3 and 3.11,

Cm = 'f2RTt/[1+(7-1)M 2/2] (3.20)

The total flux arriving at point P from the breakdown surface is foundby integrating equation 3.17 over the entire solid angle df2 within which

the surface F appears from P.From equations 3.12 and 3.17-3.20, the flux emanating from a small

surface element is a function only of local Mach number and the angleoff- the mean flow direction, #. The flux is a maximum in the directionof u , decreasing rapidly as # increases. From equation 3.17, values of(djp/dQ) were calculated for various Mach numbers as a function of 4.Results are shown in Figure 3.2. Values along the ordinate areflux#,M/flux#=0,M=3. i.e., the ratio of flux directed at an anglefrom the mean flow direction to the magnitude of flux at 4 = 00,

M = 3.0. For example, for local flow at Mach 3.0, flux at a back flowangle of 180 degrees is approximately seven orders of magnitude less

than flux in the stream flow direction. Similarly, at Mach 14.5, flux

directly opposing the stream flow is nearly eighty orders of magnitude

below the maximum value.

D. DETERMINATION OF BREAKDOWN SURFACE

A parameter for the determination of continuum flow breakdown is

given in equation 2.1. Falcovitz [Ref. 17, Appendix B] evaluates the

breakdown parameter P along streamlines of a centered rarefaction fan.

A non-dimensional gradient is defined as

Xc(M) L d1(3.21)

whose value for a Prandtl-Meyer fan is

c(M) -2-,/'/M(+I) (3.22)

19

% 0

40

0 L

<0

0 u

* ~0

c IC

00

0~~ ~ ~ 01 z c 1 c 9

200

Z) %

Bird [Ref. 18] gives the following relations for the average thermal

speed c, mean free path X, and collision frequency v:

= 2Cm/V/ (3.23)

X= (,V/_7r62n) - I (3.24)

v / (3.25)

Substitution of equations 3.5, 3.11, and 3.22-3.25 into equation 2.1

yields

P=2(y+1)r62 n (3.26)

Substitution of equation 3.18 into equation 3.26 gives

V/(y/r)(M2-1)[ 1+(/-1)M/2] '/(7 - )P =

2(y+1)r(p2t(N/j) (3.27)

To determine the location of the breakdown surface, P is set equal

to 0.05 in equation 3.27. We may now solve explicitly for the distance

r of the breakdown surface from the vertex of the expansion fan. As a

function of local Mach number,

r"=10V/(-Y/7T)(M2-1)[1+(7-1_)M2/2]1 / (-Y- 1)

(Y+l)6 2pt(N/,1) (3.28)

From evaluation of the PrandtI-Meyer function for the flow fields under

consideration, it is found that the maximum Mach numbers reached in

the expansion to the ambient pressure of low earth orbit are well in

excess of one hundred. However, the molecular back flow from areas

along the breakdown surface with flow speeds of this magnitude is

negligible. Thus, for efficiency of computation, it is necessary to esti-

mate the maximum Mach number Mm. 1 which will bear significantly in the

21

..................................................."...,.................--.-.....

present analysis. An estimate of M...= 15.0 was arrived at in the

following manner. For the gas under consideration, the Prandtl-Meyer

function (equation 3.9) was evaluated for M = 15.0. For flow exiting

the nozzle at Me= 3.0, the angle through which the flow is turned

(equation 3.8) is approximately 500. Therefore, the back flow angle

for molecules from this flow to reach a point in the plane parallel to the

spacecraft surface is approximately 400 off the mean flow direction.

Evaluation of equation 3.16 with the appropriate values gives

Iogl°(flux,= 40°/fIuxJ= 0 °)M = 1s.0 -30.0 (3.29)

That is, the flux directed toward a plane parallel to the spacecraft

surface from flow at Mach 15.0 is thirty orders of magnitude less than

the flux in the mean flow direction. Moreover, the magnitude of flux

arriving at the spacecraft surface will be less than this value. Thus

M = 15.0 is used as an a priori estimate of M=... Subsequent calcula-

tions confirm that loss of accuracy is not incurred using this estimate.

E. BACK FLOW GEOMETRY

Molecular back flow geometry is presented in Figure 3.3.

Two-dimensional projections onto the x-y plane and y-z plane are shown

in Figures 3.4 and 3.5. From Phillips [Ref. 19], the angle 0 between

the mean flow stream and a point P on the spacecraft surface is given

by

COS OU = (3.30)

where L is the vector from a point B on the breakdown surface to a

point P on the spacecraft. The magnitude of u is equal to the product

of local Mach number and sonic speed. Define the vector from the

origin to P as P, the vector from the origin to B as B, and unit

vectors in the x, y, and z directions as x, y, and z.

22

~. . . . . .... ,.-.... ......... - ,

I RI

S... . . . . .. . . . . .. . . . .

Figure~ ~~ 3. BakFo'emtr:xypas

-.- , 23

RI

Figure 3.5 Back Flow Geometry: y-z plane.

From Figures 3.3, 3.4, and 3.5, the following relations hold:

P=x R19 (3.31)

B (X1 + r cos ft) + ((R1 + r sin #)sin 0I9

+ Ri1 + r sin M)cos q5]z (3.32)

(u cos 0)^ + (u sin 0 sin 0)4 + (u sin 0 cos c) (3.33)

L LP-B = (x -XI - rCOS)x^+ [R1 -(R1+ rsin )sin~]

[ (R1 + r sin 6)9cos o'kJ(334

It (x - X1 - r COS#2 + [(R1 - (R1 + r sin j6)sin 0,]2

+ [(RI + r sin j6)cos 0]J2112 (.5

24

Hence,

Cos Ip = )ucos 0)x- X1 - r cos 16) + (u sin a sin ~

[R1 - (R1 + r sin #)sin 0]- (u sin 0)(R1 + r sin #)cos 2 0jIiiI(3.36)

The solid angle d2 within which an element of the breakdownsurface appears from the point P is

dfl -L"- Ada)/ L13 3.7

where A is the unit vector normal to the breakdown surface and do isan element of area on the breakdown surface. Phillips [Ref. 20] gives

nda =i drd (3.38)

where r is the vector along the Mach wave for a specified Mach numberto the point B on the breakdown surface (Figure 2.4). The term drdenotes an incremental distance along the breakdown surface. Withreference to the coordinate system of Figures 3.3, 3.4, and 3.5:

i (r COS j) + (r sin # i 9 sin si o ~ (3.39)

=(cos P + (sin P~ sin O)y + (sin P cos 0)z (3.40)

g=(r sin Pcos ~(r sin .6sin )A (3.41)

-' 25

- ° , °- - . t -o •- *-~. -

4

Thus,

Ada = [(-r sin2 ) + (r sin P cos P sin O)9

+ (r sin P cos 9 cos 0)^] drdo (3.42)

and

L.nd =)(x - Xl - r cos .8)(-r sin2 p) + (r sin p cos p sin .)

[R1 - (RI + r sin #)sin €] - (r sin P cos cos

(R1 + r sin #)cos 01 drdo (3.43)

The solid angle d2 (equation 3.37) may now be determined, and theflux from an element of breakdown surface to an element of area near

point P (equation 3.17) calculated. The total flux is('/1rmoxdi 2 / ncme -U2 [(+u2cos2jp)/2+(VN' /2)

min rmin

(u cos p)(eU2cos 2 %)erfc(-u cos lp)(u2cos2 + 3/2)] dO (3.44)

Note that flow symmetry with respect to the y-axis requires integration

only from 0= *ml to 0 = 900. Twice this result gives the value for

total flux.

F. SPATIAL INTEGRATION OF FLUX

The independent variable used in calculations thus far is localMach number. For given stagnation parameters and exit plane Mach

number, the continuum flow breakdown surface and all pertinent flow

properties may be determined as has been shown. The FORTRAN

program 'FLUX' (Appendix C) was written to perform the calculations

presented herein. In the program, equation 3.44 is evaluated over the

extent of the breakdown surface contributing flux to a given target

area on the surface of the spacecraft. Only areas of the breakdown

surface whose y-components are greater than the spacecraft radius RI

26

"" ... ...... .-.....- - ..... . .. '.'..'. .- .- ,'.-. '.. '' ", . . ..2''''.: ''' " "''""' '''. -.: - -

in Figure 3.5 will contribute flux to a line of points along the space-craft surface defined by y = R1, z = 0. This constraint may be

expressed

(RI + r sin #)sin 0 > R1 (3.45)

Having chosen the value of M.. 3 , a maximum value for distance fromthe nozzle lip r... is defined by equation 3.28. This value will in turndefine a minimum value of * for the integration procedure, given by

rnin = sin-'[R1/(R1 + rmax)] (3.46)

Included in Appendix C is a flow chart of the program 'FLUX'.The integration algorithm begins with the input of constants, gas prop-

erties, and spacecraft geometry. An exit Mach number, maximum Mach

number, and an incremental step value dM are chosen, and a string oftarget points P are selected on the surface of the spacecraft along the

line y = R1, z = 0.Calculations begin for a target point near the nozzle lip. In order

to avoid the singularity at the nozzle lip, the value of M is initializedas (Mo*dM) in a plane near * = *mln. The *-plane in which integration

begins is defined by values assigned to 0,, and do. In this constant-0

plane, Mach number is incremented by dM and the constraint of rela-tion 3.45 is checked. When a value of r is reached such that rela-

tion 3.45 is satisfied, the location of the point B, pertinent flowparameters, the back flow angle *, and the incremental flux from Btoward the target point are calculated. The value of M is incremented

and the process is iterated.

A flux magnitude of one molecule per square meter per second

arriving in the vicinity of a given target point is used as the minimum

value for which calculations proceed in a constant-0 plane.

f4

27

-. *Y

* I ! - .,, ,tiL,. j..S... .. ,.4.- - --... ...-.. - - "_ _

;I

That is, if

(dip)M.0 < 1 per m 2 per sec (3.47)

then the value of 0 is incremented by do, M is reset to M , and itera-tion begins in a new plane of constant-0. This process continuesthrough o = 900. Total incident flux near a given target point is equalto twice the sum of the incremental flux values thus calculated. A newtarget point is selected and the entire procedure is repeated.

28

.............5 .. . .a S 4S

IV. RESULTS

Calculations were performed as described above for exit Machnumbers of 3.0, 4.0, and 5.0, and exhaust plume stagnation densitiesof 0.01 and 0.1 kg/M. A constant stagnation temperature of 2200 Kwas used in all calculations. Table 1 indicates some of the limitingvalues calculated for the stated initial conditions. Maximum andminimum distances of the breakdown surface from the nozzle lip aregiven by r... and r.,.. M..*is that Mach number along the breakdown

surface at which the flux to any target area on the surface becomesless than one molecule per square meter per second. Note that themaximum Mach number for which this constraint is reached is M = 12.8.Therefore, the a priori estimate of M = 15.0 did not limit the accuracyof back flow calculations. The two right-hand columns give the meanfree path ) at the nozzle exit plane and the ratio of r.,. to X. Thelatter parameter gives a relative indication of the "closest point ofapproach" of the breakdown surface to the singularity at the nozzle lip.

TABLE 1

Exit Plane and Breakdown Surface Parameters.

Me Pt, kg/rnm3 rmax, cm rminscm Mmax X, cm rmin/X

3.0 .01 8 :8 1730 11.65 .0047 37.13.0 1 0173 12.80 .0005 371

4.0 .01 27 :5592 9.35 .0111 50.24.0 .10 3 0559 9.85 .0011 50.2

5.0 0 1 48 8.60 .0236 635.0 .1 2.29 .1488 9.00 .0024 63:8

The most salient data presented in Table 1 are the values for r ,

which in essence define the extent of the breakdown surface contrib-uting flux to the surface of the spacecraft. Values for r are gener-ally less than one meter. Moreover, for the cases of higher stagnation

29q.'' -; l ,', -; l : _ ' ' ? ' : : - * ' ' : " "::::::::::".' ; '' : :,.:?--

densities and exit Mach numbers, the total significant efflux from thebreakdown surface toward the spacecraft originates from within only afew centimeters from the nozzle lip.

Figures 4.1, 4.2, and 4.3 show the magnitude of flux incident on

the spacecraft surface for exit Mach numbers of 3.0, 4.0, and 5.0,respectively. In each figure, values along the abscissa (distance from

the nozzle lip) range from one millimeter to ten meters. For each set ofinitial conditions considered, flux in the plane of the nozzle exitreaches a maximum near the nozzle lip, generally well within ten centi-meters. The variance of flux within this distance of the lip is not more

than about one order of magnitude. Beyond, incident flux to thespacecraft decreases rapidly; at a point ten meters from the nozzle exitthe back flow is generally three to four decades less than the maximum

values.

The magnitude of molecular back flow is a strong function of exitplane Mach number, exhibiting inverse proportionality. The variance ofback flow with stagnation density is less pronounced. Near the nozzle

lip, the spacecraft receives a slightly greater molecular flux from thehigher-stagnation density exhaust plume. However, at distances

greater than one. meter from the nozzle lip, flux received by the space-craft is virtually independent of stagnation density, and becomes a

function only of exit Mach number.

Figure 4.4 depicts the sensitivity of flux magnitude to assumed

values of the breakdown parameter P. As P is modified, the generallocus of the breakdown surface changes as shown in Figure 2.3.Maximum flux magnitude appears to be insensitive to increasing valuesof P. However, changes in the shape of the flux distribution curve do

occur. As the assumed value of P increases, flux near the nozzle lip isincreased and the point of maximum flux moves inward toward thenozzle lip. Beyond the point of maximum flux, though, the decrease inback flow to the surface of the spacecraft decays more rapidly for

higher values of P.

30

Calculations were also performed to assess the validity of approxi-

mating the breakdown surface by the expansion wave corresponding to

M.OI, as shown in Figure 2.4. From results shown in Figure 4.5, itappears that little loss of accuracy is incurred in making this

assumption.

31

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36

V. CONCLUSIONS

The data presented in Figures 4.1, 4.2, and 4.3, and Table 1indicate that for both effusion and reception of molecular flux, theregion in the immediate vicinity bf the nozzle lip is of critical impor-tance. Therefore, efforts to minimize the molecular back flow to thespacecraft surface warrant attention to this region. In theory, ratherstraightforward and uninvolved modifications to the spacecraft structurenear the exhaust nozzle would significantly decrease incident back flow.Comparison of Figures 4.1, 4.2, and 4.3 shows that minimization ofback flow is contingent upon raising the exit Mach number of theexhaust jet to a value as high as feasible. With all other parametersconstant, increasing the exit Mach number from 3.0 to 5.0 decreasesflux to any given area on the spacecraft surface by approximately fiveorders of magnitude.

Figure 5.1 Exhaust Nozzle Modification.

Shown in Figure 5.1 is the addition of a rim located at either edgeof the nozzle, encircling the spacecraft. Benefits of such an

37

S., .... . ..,"A% . .**t*.*

*k. -7- r r- r r -

arrangement would be twofold. Most importantly, the effective nozzleexpansion ratio is increased, thereby increasing exit Mach number.Additionally, the back flow angle for molecular flux to reach a given

area on the spacecraft is greater than that for the design in which the

nozzle exit plane is flush with the spacecraft surface.

Figure 5.2 Exhaust Nozzle Modification.

Small vertical barriers encircling the spacecraft near the nozzle lipas shown in Figure 5.2 would also be effective in significantly reducingincident flux. Added without regard to increasing area ratio or exit

Mach number, the "fences" would act simply to block a high percentageof the molecular effusion from the breakdown surface. In light of thevalues of rm. given in Table 1, it is conceivable that such a modifica-tion, even if only a few centimeters in height, could be extremely

effective.

38

V7..-.. ., . .V.7Y .' •-" ' .

VI. RECOMMENDATIONS

One of the limitations of the foregoing analysis is that physical and

thermodynamic properties of the exhaust plume are taken to be weighted

averages of the properties of" the constituent molecular species.

Realistically, the number density, mean free path, and collision rates of

each molecular species will not necessarily be equal to one another at

any specified point in the flow field.

Further, in this analysis, the effect on the exhaust plume of the

atmosphere through which the spacecraft is traveling has been

neglected. Although the molecular number densities associated with low

earth orbit altitudes are small, the velocity of these molecules relative

to a spacecraft advancing in orbit is significant. The relative approach

of these atmospheric molecules toward the spacecraft is parallel to the

plane of the nozzle exit and therefore nearly perpendicular to the locus

of points describing the breakdown surface of the exhaust plume.

Collisions between free stream molecules and exhaust gas molecules

occurring in the transition region of the exhaust plume will result in a

transfer of energy to molecules in this region.

The resolution of each of the limitations above will further improve

the accuracy of the results presented in this analysis.

o39.6

.o5-

. . . . ..,". .. .

APPENDIX AON THE GRADIENT IN CENTERED RAREFACTION FAN

Joseph Falcovitz

Feb. 19, 1985

Purpose: Evaluate the Bird/Knudsen number

P -Pai ds

l d•

along streamlines of centered rarefaction fan.

Reference: J.A. Owczarek, "Fundamentals of Gas Dynamics"(1) Mach wave: sin pi = 1/M

(2) + e 0; = streamline angle

For a centered fan, flow field is function of 1.

Denote: r = distance from fan vertexs = coordinate along streamline

p = density

P, = stagnation density

Then (dp/ds) is a (-sin v) component ofr dfl

(3) d_ zs sin g =- dpds r d/f rM d/3

Denote the non-dimensional gradient:

(4) K (M) = d"

P f

40

. .. .. ................... "".""' %-~~.........:... ..." , '- 1"" .... lI

Then for a centered fan:

(5) 1(M =pd 1 +j_

Mp di Mp dki ;Ml Mp dM dMd

1 1/

*dM M [1 + (7-1)M2/21

-2M2~1)f(1+6)M /2

d1= d p[ + 1 M 2 _111 (_ _) 2

=-pM[1+(Y_1)M2 /21-

=-M[1+6'-1)M 2/21 -1p dM

(6) K(M) 2 2'T/M(Y+l)

41

APPENDIX BEVALUATION OF INTEGRALS

Given

(dIP/d)f nc -/C3 (2CUC2 I

Via the method of Noeller [Ref. 16], the substitution k c -u Cos* is made. The right-hand side of the equation becomes

nl..Cm 7T J / o (k+u cos ip) e-k e-u(Cs dk-U Cos

which in expanded form equals00 0

fTCm Teue2 CO: 2 k3e-k 2 dk + 3 u cos ~'J k2e-k2 dk

-UCos* u Cosv000

+ 3 U2 C0S2 # f ke-k 2 dk + u3 cos3 e-k dk

The evaluation of these four integrals is as follows:

1.f0 k3,-k2 dk =(U 2 COS 2 #' eu 2 C0! 2 #~ + e0 U2 COS2 f/

* 00

2. J k CoskI dk =-(u cos -0 eu2 CO 2 )/2 + (y/AT/4)erfc(- u cos

3. f0 ek2dk =(e-u 2 C03 2 ,/

-U C03 JP

d 42

00

4.f dk =(N4"T2)erfc(- u cos i'

-U Cos

where erfc is the complementary error function. Substitution yields

dip = fTCMT-2 e u2[(1+u2COS2*)/2+(.v/7-/2)(U COS

(euo5 1)erfc(-u cos jp)(U2COS 2 f, + 3/2)] dfl

43

APPENDIX C

FORTRAN PROGRAM 'FLUX'

Initialize constants,gas parameters,

spacec raft geometry

Initialize M., M.,dM.

DCn r., ntilz

Seec are pi. 1

Setsumof44u

z~ ~ ~~~~~e M Me + ;~ d ....... > ;**-*

I Ira

Is flux

from 8 to yesgreater thanone molecult

per motorper sec,

no

Set # # + d#.

noIS 0.1.?

yesTotal flux at Pi twice sum Of =flux-

Print location of Piflux at P,

,_no S e I ep

finXta@ target point,point?

'o-s P ot

no

Select now

Stop

45

LIST OF SYMBOLS IN PROGRAM "FLUX"

Input Parameters

Symbol Name Units Description

GAMMA T average specific heat ratioMW .4C kg/kmole average molecular weight

DELTA a m average molecular diameterRI m spacecraft radiusXI m nozzle half-widthME Me exit plane Mach number

DM dM incremental Mach number stepMMAX Mf ex estimate of maximum flow field

Mach numberTSTG Tt K stagnation temperature of

exhaust plumeDSTG P0 kg/m stagnation density ofexhaust plume

P P breakdown parameterIPHI, number of #-divisions forPHIDIV integrationI PTS number of target pointsXMIN m x-coordinate of,.target point

nearest nozzle lipXMAX m x-coordinate of target point

furthest from nozzle lip'

"* Other Parameters

Symbol Name Units Description

PI it 3.1415926

E e 2.7182818

RU universal J/mole, K 8314.41gas constant

NO Avogadro's molecules 6.022045 x 10"numoer per kmole

46

' 48

"-... . . . . . "

SIGMA a an collision cross section

RG R J/kg.K R = j VAXDIV variables usedLRAT'lO in determinationXCOORL, of intermediateMM target pointsC1 2/(T - 1)C2 (I + U)/( - 1)

4.C3 (T- 1)1(3+ 1)

C4 ( /C5(/1)^C6 1/CT - 1)

NUl V radians Prandtl-Meyer function atnozzle exit

RMAX rm.ex m value of r corresponding to M...

PHIMIN #min radians minimum value of 0 used inintegration procedure

DPHI do radians incremental #-step

PHI radians spatial coordinateFLUX ip molec!1!es flux arrivi in vintyo

per m per sec target pointqivcntyo

NIL M local flow Mach numberNU V radians Prandtl-Meyer functionTHETA 8 radians angle between spacecraft

surface and flow streamlineBETA 0 radians angle between spacecraft

surface and Prandtl-Meyer waveR r m distance from nozzle lipDR dr In incremental distance stepMIPS cm rn/sec most probable t~ermal molecular

speed, (2RTP"UMAG I -U' magnitude of stream

flow velocity

LDOTU L, u variables defined byLMAG ILl equations 3.33, 3.34,LNDS L.A do 3.35, and 3.43.PSI radians flux back flow angle

DOMEGA dQ steradians solid angle subtended byportion of breakdown surface

47

.....................................

:kii,.~ ~~ Z...'Z 2 . ** ** * *

-. U u dimninless flow speed,(/)'M

C7 u Cos*C8 u2 aCos 2 *C9 au

CIO eu aCos' 2

ND nr molecules/n 3 number density

DFLUX djP molecules incremental flux directedper m per sec toward target point

TFLUX jPmoleculfs total flux incident nearJpper cm per sec target point

MUIM MwMOR maximum stream Mach numberfor which breakdownis significant

48

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,.0~-)d1~~H-A.740% C4d- 94 H +- 'I -4-i + C

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Cn H * - r,%~ C I) WIS *1 it %-^- pk%*P a *0 . . (AZL)Cntnw ~ 1 Pd1Tc .000.4 *f It 1 110U' 11 11 11 11 11 0~rH4QZ - q*I 4-

Pk-9HZ ItJ C-*1 A Hgr-4 HHWWU0=0.3wwH it it i ~if 9 i C) 94 it i >1 0 Hinz >4 40H i 1-(-I-1111 It 11 Of

it04~) F-~ P0 H 21 4H ~ 1-90 9HZ

OHO H-4 0 HZP0.. ,. H O r

PW41. H Wa kUV)

49

r4

C4

%-Id * 4*%-0 Ed -

* ~Ea~~ W

+4 -- Af

A. U-0

20 (n*E'- 0 *~-4r- 1i'l E- HW4

CA W4' .4 + 'r-4

0 H9 *:D '- - * ~-41 cH-D W-X H) H 0f U - * d'

X~E-4 N-' OH -4- '-.4- +uln >4- 00 4 O-~4 4 H'-Or-4P

- W god'-" aw cN2- IM 'T =409 +E -44C *. *I-44* % -- 4* -- H O U 0ZU'-"4

Um t"4CH 94* U o=9 Ed * 9~a- P~L

2'- r-I -%O ;z:A4 -i 2 Pr0- %~fe' 4 H- -* 04*OH -4 N ,-EI N HWa N 0 0p F-4 p~x 'UC'-4 '

PP4 H ~H* ROZ-4'-,-+ w-4 hL)P4U-f-'O'Ir f-4u24 N-0 ~ Hf4 ~O - -- * + + 4 CO,4* =:v.4-UCI)ow& 0 L)f %U-o Ir4

*4) H-A~. 4 H =ZEZ *4wz-.u-"KO'.-- 4 )H-'-a- 04 -- -C I- U:DO4 HZ=Z-Z JU0p - E-4-C4 4E-.4=Ht

NZ -N P-- d** E4W c

- 4HA - ..g 2z-!44.cU E-4- J - '4CI -.- ''. IJdH* iLOO4H 2P' ZEd .- '- .. I-O ~ OS f-04* -z ."*4

Q , 0l~~ *H'-4 + W<, -O + HOOP--.,fi C1 4 -i - ~ N -4014H- If4 Z. 4U 4 I E4P 4-"H

PH-O "t-I 4a4 4a--A4 1 H 2p +OI-4 f-.Edf-sj4HL)~ 'f-c~-

OW-NNH* - ~ U4 AMP. UOH "IL)*4.Oii -4dH-40 HO-4-0. II F P0-30 MI W% HE*~4 o'-2 -> C3 %-0 4 ,.~~ Ni of44,- 4* '-WOWNa H-I-4 lii -

-4 -I~'-I fa2>IiGJ of "0 04~I fl-vK * - - '- 1E42Ed~ iEd-4d d *HH NICIJ0 O-ViEd-I 4 -HHW '--"

*UH-4"o X H-"H ee-44 C4%--4

4 0 041 P-3 cnEd 1- EdHU 0 'OU rlU2UH6

U) U U UU) UU. LU

50

MC.-I 4 Q CjZ.4**Hj N~- --

m~- WOO ~ 0-000

*'44 zoi 1-4 1H-4

-t0 (nQ.O 0 E- 0 -__t, A. 4 1 4 H

-- 1H UO~ - ZH 1-I

HE--"

0. '-4.- -1*-e 0L'N 049QI-4 m II-4-4C 1: Hw:r-4 *d4 -- H-4:~ z nU

* P4 4 HU X OZoH ,-3 H

H N - 0 4x:U + -4 f I%..0 Pa'-E- *o 1:4~U 004 4* v*-. C-44 NA 0Id- c .li 44 4*0-- it 5a *: 4 H P4OH4 HtE* -4 -1WU-0 U - -'. 4~

OX*-- --0' U %0 ZOO X' 0 tI- -04H rzo E-lUE-4 N- -403'-O H ' ~ J O

P3~~ O4*-'~ HL)~ -K %~~d~ -# EH -- aE-5-

p4- C3-I%#L"-- 03cf' qc 0i v-4 HU3 X4. *'Ooz1: 0-H14-4CU § -C* E r-4 0 W ~~ 0NOHc ' -Oi- .UHO O I-+1 .40* H ,3HP44"1-~'-- 9ctE4HV *-.-*-e4 A44 -0 P4P OH-.s" H

0 -4 UQ- H*))-4~U H-P4'.fl *Xr-4 w -0 - *p oo *- x--CC)U ~-fH iiII24H: - 0-I - -0H * .1-43 O M 0 :41 --

-K P,-' ' P404~ 1:Jl4-3-.W HZ.. r444~--M-4% 0*. -%M Z4i U:JP4 ~z *i *,..34 E-- =H 5- X'-'-4 -Z'- H 4<H P--< p)0C - - - :!'-' - p NEO-4 CN4 XM MO -0W PJH-44: :-' PEA H:3Er---r"-O-4-4Z-- '-4:3:) II0 1-0-54~

E-4-~~- PN C'4x--- N4 *'uwA r;, H9z~~~~ ~ ~ 0+AQHP.Co=w0x<1m0 oif "1-nO-O-O

CM- Lr-Z~cJ. P4-,OcPI Z N~n-= H + 1:3*.~o .0 N - --Cnt4 ~ --XS4 UUI- U-r H.e-C%0 " O 04

CIO f-4:)P4"- rZHZ Cn51

LIST OF REFERENCES

1. Bird G.A., "Direct Simulation and the Boltzmann Equation",The Physics of Fluids, vol. 13, no. 11, 1970, p. 2678.

2. Shidlovskiv, V.P., Introduction to Dynamics of Rarefied Gases,Elsevier, 167, p. 15.

3. Aerospace Research Laboratories Report ARL 73-0134 AState-of-the-Art Survey of the Mechanics of Rarefied Gas Flows,by K.S. Nagaraja, September 1973, p. 48.

4. Patterson, G.N., "A Synthetic View of the Mechanics of RarefiedGases, AIAA Journal, vol. 3, no. 4, April 1965, p. 584.

5. Bird, G.A., Molecular Gas Dynamics, Clarendon Press, 1976,p. 5 4 .

6. Keller, J.B., ,,"On the Solution of the Boltzmann Eguation forRarefied Gases , Comm. on Appl. Math., vol. 1, no. 3, 1948.

'.7. O, cit., "Direct Simulation and the Boltzmann Equation", p

8. Bird, G.A., Breakdown of Continuum Flow in Freejets andRocket Plumes, paper presented at Twelfth InternationalSymposium on Rarefied Gas Dynamics, Charlottesville, Virginia,7-11 July 1980.

9. Ibid.

10. Air Force Weapons Laboratories Technical ReportAFWL-TR-72-28, Hydro en Fluoride Laser Technology Study, byF. Mastrup, et al, Octoer, 1972, p. 91.

11. Williams, R.A., Handbook of the Atomic Elements, PhilosophicalLibrary, 1970, p. 42.

12. National Aeronautics and Space Administration SP-105, VacuumTechnology and Space Simulation, by D.J. Santeler, et al, 1966,p. 5.

13. Gordon, A J and Ford, R.A The Chemist's Companion AHandbook Pf l.ractical Data, Techniques, and References, WileyInterscience, 1972.

14. Owczarek, J.A., Fundamentals of Gas Dynamics, InternationalTextbook Company, 1964, pp. 153-159.

52

'.': "~................... :,.

15. Ibid., pp. 442-443.

16. Noeller, H.G., "Approximate Calculation of Expansion of Gasfrom Nozzles into High Vacuum", The Journal of Vacuum Scienceand Technology, vol. 3, no. 4, 1966, p. 203.

17. Falcovitz, J., On the Gradient in Centered Rarefaction Fan,personal notes, 19 February 1985. (Appendix A)

18. Op. cit., Molecular Gas Dynamics, pp. 6, 59, 70.

19. Phillips, H.B., Vector Analysis, John Wiley and Sons, Inc. 1933,p. 7 .

20. Ibid., p. 94.

..

,1.

'.

I..

J

"53

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22304-6145

2. Librarv 2Code (142Naval Post raduate SchoolMonterey, "California 93943-5100

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4. Distinguished Professor Allen E. Fuhs 4Department of AeronauticsCode 67FuNaval Postgraduate SchoolMonterey, "California 93943-5100

5. LtCol. Doualas Kline 2Code SDIOjrDEStrategic Defense Initiatiye OrganizationWashington, D.C. 20301-/I0

6. Commandina Officer 4Attn: Lcd-. Scott E. McCartyFighter Squadron 101N.A.S. OceanaVirginia Beach, Virginia 23460

* 54

0%

'. h.. * o* % *