AO-A115 367 FOREIGN TECHNOLOGY DIV WRIGHTPATTERSON APR ON FIG 20/4THE NUMERICAL SOLUTION OF MACH-ZEHNDER INTERFEROGRAM ANALYSIS O-ETC(U)APR 62 P DING, S PAN

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THE NUMERICAL SOLUTION OF MACH-ZEHNDER INTERFEROGRAMANALYSIS OF SUPERSONIC AIRFLOW ABOUT CYLINDRICAL AND

SYMMETRICAL PROJECTILES

by

Ding Peizhu and Pan Shoufu

DTICSELECTEJUN 10 19823

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THE NUMERICAL SOLUTION OF MACH-ZEHNDER INTERFEROGRAMANALYSIS OF SUPERSONIC AIRFLOW ABOUT CYLINDRICAL ANDSYMMETRICAL PROJECTILES

By: / ing Peizhu and/an Shoufu

English pages: 16

Source: Acta Scientiarum Naturalium UniversitatisJulinensis, Nr. 3, 1981, pp. 60-71

Country of origin: ChinaTranslated by: LEO KANNER ASSOCIATES

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THE NUMERICAL SOLUTION OF MACH-ZEHNDER INTERFEROGRAM ANALYSIS OF SUPERSONIC

AIRFLOW ABOUT CYLINDRICAL AND SYMMETRICAL PROJECTILES

Ding Peizhu and Pan Shoufu

SU~Th1ARY:

In this study, four physical conclusions and the numerical solutions

described in [1] concerning the Abel transformation in Plasma Spectroscopy

involving shock-waves extended to the Mlach-Zehnder interferogram analysis

in supersonic airflow about cylindrical symmetrical projectiles. Regarding

the relation between the interferonetric fringe shift 6(x,z) and the

concentration variation p(r,z)- Po obtained at each point of the supersonic

airflow field, we obtained three physical conclusions similar to those

described in [1]. In addition we provided the interferometric fringe shift

5(x,:) obtained from observations and the coefficient formulas of the entire

transformation necessary to calculate the concentration variation p(r,z) -p0

at every point of the airflow field.

We borrowed from source [2] the airflow field density computation

method provided for the Mach-Zehnder interferogram applied to supersonic

symmetrical projectiles, and obtained satisfactory results. Also, a

simplified method for computation is given for the case of additional

shock-waves present in the airflow field.

I. INTRODUCTION

The question of the airflow about supersonic cylindrical symmetrical

projectiles is old and of sizeable importance. The Mach-Zehnder interferogram

analysis (*) has been for many years an important and practical research

tool. Since the invention of digital computers, extensive applications

occurred in the M-Z interferogram analysis. The method of interferogram

analysis consists in measurements of the interferometric fringe shift

6(x,z), the research of the concentration variations p(r,z)-p at every0

point of the airflow, using the Abel transformation; thereby we obtained

the curve of density in plasma spectroscopy.

(*) below: "'-Z interferogram" or "interferogram".

1. U

In the past scientists have applied the integral of Stieljes to

obtain numerical solutions of the above problem [3]. But experimenters

are still not familiar with the application of the numerical solutions of

the integral of Stieljes; further, the results computed thereby present some

ambiguities in this case of shock-waves. In [1], we discussed some

characteristics relative to the usual simplified solution of the Abel

equation. By. using the Abel equation with a radiation frequency I(x) in

plasma spectroscopy involving shock-waves and an emission coefficient (E(r),

we demonstrated that E(r) can be computed from the usual Abel transformation,

and we obtained the formula E(r) for the computation of the amplitude of

the shock-waves.

This is why we suggested in [1] a new computation of the Abel

transformation in plasma spectroscopy involving shock-waves.

In the M-Z interferogram analysis, equations noted as 6(x) and p(r)-p o p

the radiation frequency in plasma spectroscopy I(x), and the emission

coefficient E(r) are mathematically identical. Therefore it is possible to

extend directly the conclusions and methods of computation obtained in [1],

to the Abel transformation of interferogram analysis about cylindrical and

symmetrical supersonic projectiles. The method of computation is simple.

The physical theory applied to the shock-waves is equally understandable.

In the second part of this study, we describe the Abel transformation

of interferogram analysis about cylindrical, symmetrical supersonic

projectiles. In addition, we extended the conclusions obtained in [1]

concerning the Abel transformation in plasma spectroscopy involving

shock-waves to the present case. In the third part we have listed numerical

solutions. In the fourth part we discuss the case of an additional

shock-wave. The fifth part is a possible plan for a simplified computation

in the case of an additional shock-wave.

II. THE ABEL TRANSFORmATION

Figure I illustrates a point-headed supersonic projectile in flight.

2

LMid

We suppose that conditions are such as: it is cylindrical and syrmetrical,

and that there are one or several tapering shock-waves roving towards the

back. Section 1-1 is illustrated on Fig. 2: the hatched circle in the

center represents the cross section of the projectile; the circumference c

represents the cross section of the shock-wave. Outside c, the airflow has

not yet been affected by the wave's interference. The density and the

index of refraction are respectively p0 and n0 ; within c, since we are

dealing with a cylindrical, symmetrical projectile, the density and the

index of refraction only reach the distance r related to the center 0, and

are respectively:

n=n(r)

SY

aY

II I ,I L

- C.-t 0 Ce

Fig. 1. Illustration of the cylindrical, Fig. 2. Illustration of photolinesyretrical supersonic projectile according to cross-sectionin flight. 1-1.

Fig. 2 illustrates the interferometric fringe shift corresponding to

section 1-1. The value 6(x) at point x is the light crossing section 1-1

refracting the superimposed effect of one beam going fror -y to yl and awa

from the y axis for each value of x.

3' "

According to the usual hypothesis, the interferometric fringe shift

6(x) and the refraction index difference n(r)-n or the density difference

o(r)-o ° satisfies the equation: (*)

3(x) = _(re(r) - ns)dy A* (p(r) - p.)dy,

in which K is the Gladstone-Dale constant, X* is the wave length in

monochromatic light and total vacuum necessary for obtaining the interferogram.2 2 2Focusing on the cylindrical characteristics and the value of x +y =r , the

above equation becomes:

,A (x) -. ( , r) -Jr :' - :' ,r. (I)

We assume that the shock-wave is a simplification of the first

category of o(r)-c (or n(r)-n ). If c is the most external shock-wave, then0 0

outside c, o(r)-co =0. And therefore Eq. (1) can be written:

& x) -- 2K ....r 4 - -r,

A. . J r -xg (2)

in which R is an arbitrary constant superior to c. Besides the constant

2K/X*, Eq. (2) and the Abel equation encountered in the study of plasma

spectroscopy involving shock-waves are similar. Bennett and his colleagues

have studied Eq. (2) in [2]; in [1] we discussed some characteristics of the

solution of the Abel equation. If we apply them to Eq. (2), we obtain:

1) on the M-Z interferogram's arbitrary cross section 1-1, the fringe

shift is continuous.

2) If, on section 1-1, the shock-wave c belongs to the first category

interrupted 0(r) and p'Cr), then inside the shock-wave we have:

2' (X) a Kx ft p ' (r) + 2Kx P(c)-P(.') - (3)

especially when x-c.

V( X Kiic- _p~c)-pc-)(4)

(*) See [2].

A4

3) If the fringe shift is sirooth outside the peaks c, then ir '

exists; when x-c, 6,(x)-- L/j then shock-waves

appear outside the airflow circle r=c, on the cross section 1-1;

the density saltatory value of the shock-waves will be

p(c.) - p(c,) .. A.L ,(5)KJ 2c

further, on [c ,C] and [c,R]0

pA) - p, - , (6)

These conclusions are also applicable in the case of a single

shock-wave.

III. NUMRICAL SOLUTIONS

We will first discuss the case of c with a single shock-wave. For a

convenient application to the case of several shock-waves, "The P(r)-Po=0

external to c" is transformed into: "on [c,R],p(r) and p'(r) are continuous

when r)R, P(r)-Po=0." Evidently, the former is a particular of the latter.

With [0,R]N, we have the following partition:

o =X, <x,< ... <c = x. <x., , < ...-<x = R

or o =X,< ... <C9 = Xw<Xw, I< ..-. = ¢<X. .." <XV R.

We also note: h=x,,,-x,,==3(x,) . According to conclusion 2

obtained in section II, the derivative 6'(x) of the interferometric fringe

shift tends toward the moment of the shock-wave with the value 1/,FI-x

and toward the infinite from inside the shock-wave. Consequently, one more

term is used in the expression of the shock-wave, 11 .c'- x , for the

breakdown of the comparable 6(x). On [x %.2 ,c] and [x.i 1 ,c] we extract

P.-.(x w>P.-, (a)i+# + .+ -r~

14 _C(7)

the comparable, 6(x), zero.

We can state P(x)=P.I,(XX)=, k=v-3,u-2,v-.V,

including a, b, c, 1= j-6.+3 .- ,-36.-,+5.-s 1 (8)

Then extracting from D=3V 2v -I -3V 4v-4 +J--6V-9

Cx,,x, ,(j=+.w+2,'",v-3)

P, (x) =a+bfx +cx" 4+d,x s +1d-4-i--xii - ,

When c0 0O, we extract from [c ,x w+]

P. (x) = a. + b.x +x+J c' - x" ,

When co=., we extract from [O,xl]

PO (x) = ae + bx + c-g ? dx" + c ed".--

The comparable interferogram fringe shift 6(x); in every srall interval

external to the shock-wave Cx,,x, (=v,v+],...,N-1) we use the

ternary or binary polynomial:

P,(x) =af +b,x +csx +djx3,

P,(x) =a, +bix +c,x2 (j=v,N- 1)

as the comparable interferometric fringe shift 6(x). See [4]. By using

again the value 6 k we can state the various coefficients ai,bi,c i . According

to the conclusions obtained in section 3

P(r) '(x) dx = -

By using P'i(x) as stated above to substitute 6'(x), we get:

"As J- dx.p(r,)-Po=- xK . Jx'-r'

After computation of each integral, we have the parallel systematization:

.- !p (r.,) = pa + E .ai ,

p(r,) =P0 +4 -., . 3 b. (j- w+ 1.w+ 2.....v-.,+ I,....N- 1);

.N., (9)

p ) 6

We will note:

2xTK Th'J 2v-I1 /. .-j B,'- J 6U-9 ,B 4 =j Ht'-16f

I H -: .+ 3 B,./ .- R (j .k) 4fk . 4 3j ) 1 (k + J s j ) - k4 k ,-

T~j~k ~~act1 arctg

+O~2 O -arc sinI +

(. y-4 -3J ul- 4 + -:jjU'- I - v) -F(o. 1) -12

-'-4- 2J +~ -u+ T(0. I) +

+ Jl k2)~ 4J + GJ v" -+64v- k2 -41 v;-(k -62

AIw - {'(w+0.5)4u- (w+2)' -2(W4- )4 vf-(wu+ I) +

.i(w+1..)4u,-w') lnw+

+ 12 IV2-W+-)')-3./'-(w4-2)9+3 4ui- (w+ l)-

-4v~-w'}F(w.w+ I) +

+ arctg v__ (w+2) +

+. I {L(d-h 2 T4/vn -k+ JPf+ Js... -

- 4;~-?~)'+. ~ ~j~).wA+ T(uw,k)l -

I1F(w,V-.2) {(v-2.5)B,-2(V-2)B.+ (v- .)B*}

7

u- +

3 . u l + D *,' ' ( j -4-1

.~V-(; j) F Fj.j)+ arctgj v- +

~( (~:~~+i 4J '-k+ )-7! +61v-k'

-4- -V I))!. ~Il+ J T 2)2)-j Fi) + (~k)

+ (v In 125 --~-f 2 +(~f J ;'J(

(,4, = C*Fw +2 w-L-- u - 3)-(2-In'

13*15 L I22 when w)

{ F (w~v +~ 2) - iF (ua 4- ) - 3f(w .:'rI 1w 1 'A~}hcn %* I

{2C10.35- 13 In? -W2ht2Men w=

-2C 1(w 0.375w- 3 n lWhen wi

C~ CF (.w 2) - i F (w~ w1 + 3F,) -F (. zv+2 - 13 F- -t x

6(2 Inw+ IW2,when~j2.4.~)

a~, = C{Fi.j+ 2)- iF(j,j+ 1) -3F(j,j) 424iljl.

a,*,*, ~ ~ ( - F jj+3 - w + j zvj 42.. , ,)+ I v.. - 2'.j.)-V C- )j

a, C*{F(j.v- 5) -,tFQi,v- 4) + 6F (j~v- 3) - 3F(j~v - 2)

4-6(2v- 3) In (u-2+ .1(v - )a) P 12J ( - 2)~ j

12C* B J' E

(j WW **L )

a, ,C*{F(j.v- 4)- 4F(j.zv- 3) +3F(jv- 2)- 2I(t- 2) < 1fl(u- 2+

J(u-2)t! j7 ) +24J Dv22 j } %{(3+ i? 1

-F-Jhj -73A(j) +3arctt B,' 2?j

C- C F(j,vl- .3) - F (j.zu,- 2) + 6(2--5) It,(v - 2 + j 1,v -2)? - j) -

-12J (v- 2)? j? -' D2 (-D-.-/? - (2) G)L(j) +G.,I(j) +

+ 3A(j) - .3arct~r 8.j - JV 04- Z

2B

a,{( 2C *1 .FV .v- ) + EJ(vt'- 'I) -3 (v- 1)- 2)ct -

(- -2 9) nt- 1) +2('- 3)n(v - 2) + 2 20 Q1 F 1 e,3

- ' )B Lv t)+ F-1 z,- .1) + An- 3)1)- arctg ) I

j 2Z' -

a. 3 -C C{F (v- 91,z)- 2) - 3F(v- 3.v - 3) - 6(2v - 7) In (v- -

-2 1(v -2)rifl(zi -2 + j2-5) )+ 2 Q 2v-

B.,-3A (v- 3) + 3arctv -- c--

,J2L-5J

~,-," 12C I- D.~.1.uE.B,.1nv+fl2.a,_2, - 12C 3+ E.. h3P3-D B 1 n V4 ,D- 2 v 2 2-2

al 2C* - 3 ir B (3 - ) 13- B O - D ( E-2 I flB

_.tl-B,(3E-2Dh 4D-3B1 + B' 3-2)I

a,..,, 2! !~j+ B. E - D -B,+ '(E - D)In± + Bl} +

+Co{F(u-2,v+2) -F(v-2,v+ 1)- 6(2v43)fln(v+B,))1

9

+ B, (3=E - D){+-3 - 3E(-EDD)B,(3E1

= I{ L, -B,(E -2D)+D- 3B, + 2 OE -2D) 'In _},

D 2 1 2 Jv-1

+C*{F(u-l,v+2)-F(v-.+)-6(2+3)n(u+B1 )};

a,,,+, =C{(F(j,u+3)-4F(j.t,+2)+3F(j,v+1)-244u'V: -4.+

a,,, CO{F (j, v +4) F (j,v+ 3) +GF (j,u +2) -3F (j. v+ 1) +12JU~p

6(2v + i) I n(v +lv2j)} (jW, W+ 1, ....,v);

'z,,~=01

u.-C*( -I'(u,u +1) +F( 0+ -) -6(2v+ 3) Inv};

a*,=CJF(j,NV-1)-4F'(j,J-3)+6F(j,N-2)-3P"(j,NV-1)+

a ~ C*F(j.N -3) - IF(j,N-2) +.3(.N- 1) -

* ~~- 2iGV- I)tif(N +4N'- *) +244i "j N~~p}( ~ + 1 ,---N -3)a ,.---Ct(-3F'(N-3,N-3 -i-6F(IV-3,N-2) -3F'(.V-3,N- 1)

-6(2N-7)Il(N-3)+6(2N-i)Il(N+46N:-9')-1246N-9 }

a-- rSC*(3F(N-2,NV-2) -3F(N-2,N-1) +6(2N-J)Il(N+2,4N -1)

-244-i-- +24(N-2)Ifl(N-2)},

U,-.-=C (3F(N-2,N-1) -3F(N-2,N-2) - 21(N- I)-

in (N + 24N-: j) + 48 4Nf--I- 6(2N - 5) in (NV -2)11

6(2N i)IN+2N- 1 12J2N-i}

N- +24 4zN i}.

According to conclusion 2 obtained in section II, and Eq. (7), it is

possible to establish the approximate rebound equality for the density of

the shock-wave.

10

P(c- P = A*1I2K, (10)

in which 1 is computed as in (8). Especially when rnc, and in the case

when p(r)-o =0c

p(c.) - p, = A1/zK. (11)

Supposing the measurements of the interferometric fringe shift give

the following readings:

((x) + I{Ci 2 !5C<<I6 W x = ff c.<.x-<RIIl.f

and if in addition f'(x) is continuous with [c OR] and f"'(x) a fractionate

succession, we have proven (in a further study on the computation of errors)

by the above calculation of the density of the airflow p(ri), the existence

of an error such as:

P(r,.K Mh- (12)

' ~~~~~~~max ''-)i aerltdanin which the constant M and c, R and m02.11, ( ) are related, and

the positive number a equivalent to the wave h, being extremely small, can

be arbitrarily small.

A (P(C - p) (c,)) ; " K M max (x) (.) 7 (13)

The constant NI' and c are related. From computation in (13) it is

obvious that the approximate calculation (10) is rather precise.

From Table 9 established in [2] regarding the interferograms of spherical

projectiles, we have estimated the values of S(x,z), the interferometric

fringe shift, and used them in the computation with the method described

above. Fig. 3 shows the isograph of the comparative density values p00

inside the shock-wave. Fig. 4 is the reproduction of Fig. 10 borrowed from

(2]. Table I gives the concentration saltatory values of the shock-wave

11

between I and V, as well as the density measurements inside the shock-wave

P0 It is apparent that our method is simple and practical.

10.6 0.7 6.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.5

/>-

'.7 .112.0 P. IM

Fig. 3Density ratio within the shock-waves PMr)l'p*

Key: (1) axial distance; (2) radial distance

0.5 1.0 ~ eu

Fig. 4. Density ratio within the shock-waves p(r),"P

Key: (1) axial distance; (2) radial distance12

Table 1.

-- o 0.61 - 0.41 , 0.25 - 0.19 0.14

1.0 1.60 1.44 1.38 1.33

IV. CASE OF SEVERAL SHOCK-WAVES

When a point headed cylindrical supersonic projectile is in flight,

two or more shock-waves can be observed. As illustrated in Fig. 1, cross

sections 2-2 and 3-3 correspond respectively to two or three shock-waves.

With two shock-waves as an example, supposing that the wave originates

outside r-cI and r= c2, the projectiles' cross-sectional radius is co:

we extract the isometric partition;

In the expression .X,..,, ,.(j~v,w+-,...,u,-1) we extract the

binary or ternary polynomial comparable 6(x) with the additional term

lJc:'x" . In the expression Cx1,xj ,, (j=u,,u, +,...,u+1 1) we extract the

binary or ternary polynomial comparable 6(x) with additional term ,4J -x*

By repeating the process described in section III, we can definitely

extract 11, 12 and the coefficients of each polynomial, introducing them in

p(ro) - pe 2 -x -QK ,, +-9-0- r

which gives us the parallel systematization:

13

p(r,) = P9 + a,,. a ,;

* -I

j(r 1 ) =p0 + ag.,,5 ,., (j~w+ I,...,v,-2,v5 + I,..-2-");

hI- I

p(r, ) = pe + ap " 00 i-1

p(rj) = ps+ i. a.& (3.kl~s- )

in which each equality of transformation coefficient a is comparable tok,k

the derivation of section III.

On the external shock-wave c, the concentration saltatory value is:

p (c,_) - P" = A0*I;/2K .

On the internal side of wave c, the value is:

p(C,_) - p. A'I+/2K.

in which:

2 -h I- - .

D, -:j2v,- I -:j, v.- I +dj;.-L "

On the internal side of shock-wave cl, from the concentration saltatory

values

andi{0.) 4- ., . , , ,

we obtain the concentration rebound values of the internal side of the

shock-wave:!'¢ ) 2K ' * : ", '' "

in which:

D 1 ( _h ., + 3 , ., _J- 6 , .+ , )o

D, =3 2u,- I - 4 ,- + - •

14

V. PLAN FOR A SIMPLIFIED COMPUTATION

We take as an example the case of two shock-waves. The partition is:

CC<X, . =c2. ... ,

Previously we only considered one shock-wave c2 . As in Eq. (9) in section

III, we extrv- t v=v-,, w=vl, to compute the value of the concentration

p .+, ., P(r,.,1)

We distinguish the values of concentration of the shock-waves c1 and c2.

The rebound value of the airflow concentration of the shock-wave c2 is:p(c,. ) - P. = A1 2 /2K,

in which 1 2 is similar to section IV. Secondly, we will review the case of

one shock-wave c1. Again according to Eq. (9), we extract v=v1 , N=v 2, to

compute the concentration value:P P(r.),. ...., P(r,.,), P(r,)

We distinguish the concentration values in the shock-wave c and

outside of it:.., P(r,_), p(¢C, )o

The airflow concentration values are:P(c,-) -,p(c,+) = A1°,/2K,

in which 1 is similar to the value obtained in section IV.

The method described above can repeatedly be applied to the case of

several shock-waves.

is

Ila Ding Peizhu, Pan Shoufu, Abel transformation of plasma

spectra involving shock waves reported at the "All-China conference

on High Energy Density Dynamics Testing Technology", Changsha,

June 1980.

;2J Bennett. F. D. et at.. J. ,ppl. Phw,.. 23. 453 (1962).C3, Ladenberg. R. D. Physical Measura,mnts in Gas Dynaauc and Conbustion, Oxford

University Press. 1955.

C43 Bockasten, K.. JOSA.. 1. 943 (1980.

Abstract

The method of uumerical solution on four physical conclusions and

-The Abel Transformation of Plasma Spectroscopy Involved Shock Wave'

obtained by the same authors in L.) is extended to Mach-Zehnder

interferogram analysis of supersonic airflow about cylindrical symmetric

projectiles. Three similar physical conclusions about the relation between

*i Ithe interferometric fringe shift, 3(x,, measured from M-Z inferferogramand the density difference, p(r,z)-p , at each point in the supersonic

airflow feild are obtained, and all transformation coefficient formulas used

to calculate the density difference, p(r,z) -p., according to the intcrfc-

rometric fringe shift, 3(x,:), have been given. As an example, by use ofexperimental data given in C2), airflow field of cylindrical s. mmvtricprojectiles is calculated and the better results have been obtained. .A

simplied calculation method which deals with many shock %aves existint,

in airflow feild has been given.

16