SELECTED TRANSLATIONS ON SOVIET ROCKET ENGINEERING

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R004200140002-4
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 71
• Document Creation Date: December 27, 2016
• Document Release Date: March 14, 2014
• Sequence Number: 2
• Publication Date: December 25, 1959
• Content Type: REPORT

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JPRS: L-1188-N
23 December 1959
SELECTED TRANSLATIONS ON
SOVIET ROCKET ENGINEERING
Distributed. by:
OFFICE OF TECHNICAL SERVICES
U. S. DEPARTMENT OF' COMMERCE
WASHINGTON 25, D. C.
IC '1 RVICE
205 EAST li2nd. STREET, SUITE 300
NEW YORK 171 N. Y.
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SELECTED.TRApS,LATIONS-
,
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? IIPRS: Lsa.188-N
CSO : 3905-N/2
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This series includes translations be .a.elected items from the
Soviet literature on hypersonic .aerodynamics0, magnetohydro- ?
dynamics; space flight mecifanicOl'prOulsion systems (liquid,
solid, nuclear, ion, plasma), proPeliants and combustion, in-
strumentation and control, guidance. .and.
and structUred, and'spade'ocitftUnicdtiOnd. The series is pub-:
lished as an aid to U. S. Government research. '
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? Page
? -, _ , ?-
On the Theory of Gad Flow in the Lay0:Between the .Surface 1
of a Shock Wave and the Dlunt'SUrfade of a TOtAing Body
(F. A. Slezkin)
Approximation Method of Calculating Shock Waves and Their
Interactions (G. M. LyAhovcet
. , .
Deceleration of a Supersonic Flow in Wind Tunnel Diffusers
(N. N. Shirokov)
Shock Tube for Measuring Drag Coefficients of Bodies in Free
Flight (Yu. A. Dunayev et al)
Information on the Status of Soviet Research on Hypersonics
(M. S. Solomonov)
Flow Around a Conic Body During Motion of a Gas With High
Supersonic Speed (A. L. Gonor)
Calculation of Axisymmetric Jet Nozzle of Least Weight
(L. Ye. Sternin)
Experimental Investigation of Self-Oscillations of Square
Plate's, in Supersonic Flow (G. N. Mikishev)
17
27
39
45
51
61
70
?sz;. ?
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Page
Self-Oscillating Systems in the Presence of Slowly Changing 78
External Influences (A. A4 Pervozvans4Y)
One Approximation Method of,Inivestigating Se1f=.0scillating 85
Systems in the Presence of Bibilly Chain g External
Influences (V. I. Sergeyev)
On the Motion of a-Siendere:$.4ka, Bo4y- Under the Action of a 90
Strong Shock Wave S."GrcgOisSrdn)
Useful Interference of an Airfoil and Fuselage in Hypersonic 94
Velocities (G.-Le Grodzovskiy)-
Flow Around Bodies by a Non-ideal Gas Flow With High Supersonic 100
Velocities (G. A. Lyubimov)
ifonlinear Problems of' .?t'at",ility -of Flat Panels at High Superr.e% 105
'sonic'Speeds (V. V. Boiotin) .
"
Supersonic Flow Around "fiat Cuasitriangular Wing, df -113
rength (P. I. ZheluddvY , -
One Form of Equations of Supersonic Gas Flow (F. S. Churikov) 117
Estimation of the Permissible Irregularity of Rotation of a 123
Reversible Table fpr Testing Floated Integrating Gyroscopes
,
for Drift (G. A.:Slomyanskiy)'
. ,
All-Union Conference on Static Stability of Turboraachinery 127
(le. I. Boldyrev)
. .
Coordination Conference on Staiility of Gas Tilrbines
(Ye. I. Boldyrev) s,
?
133
-0-1108
ON THE THEORY OF GAS FLOW IN Tah LAYER BETWEEN THE
WRFACE OF A SHOOK WAVE AND TEE BLUNT SURFACE OF A
ROTATING BODY
N. A. Slezkin
deleniye Tekhnicheskikh Nauk,
Mekhanika i Mashinostroyeniye, [News
of the Academy of Sciences USSR, Dep-
artment of Technical Sciences, Mechanics
and Machine Buildingj, No. 2, Mar-Apr 1959,
Moscow, pages 3-12
As is known, when a body moves in air with a velocity exceeding
the velocity of sound, a shock wave is produced. If the forward portion
of the surface of the body is blunted, then the surface of the shock
wave is located in its forward portion at a small distance from the sur-
face of the body. In 1946, we proposed in one of our articles [1] to
consider this intermediate layer between the surface of the shoe:: wave
and the surface of the blunting of the body as a Reynolds layer, i.e.,
as a layer in which the flow of Jar; is affected essentialij by the pres-
sure and by viscosity forces. This assumption can be justifi2d in the
followine manner. It is known that the influence of viscosity is of
importance nOt only for the flow of as near solid walls, but also for
the flow within the limits of the pressure jump itself. As long as the
considered intermediate layer is bounded on one hand by a solid wall and
on the other hand by the surface of the pressure peak, then the viscosi-
ty of the gas shoeld exert a substantial influence on the flow of gas
in such a layer. Therefore, the problem may concern merely whether the
viscosity should be computed in accordance with the Prandtl-layer model
or in accordance with the Reynolds-layer model. In our second article
[2] we have shown that the Reynolds equations, which he proposed in the
approximate hydrodynamic theory of flow in a lubricating layer [3], are
applicable not only for small Reynolds numbers, but also Reynolds numbers
on the order of g-11 where is the ratio 'of the mean thickness of the
layer to the length of the longitudinal extent of the layer. On the o-
ther hand, the Prandtl equations for the boundary layer are correct for
Reynolds numbers having an order e-2. In both cases the characteristic
dimension of length, 2, is taken to be the length of the longitudinal
extent of the layer, and the characteristic velocity is taken to be the
maximum value of the modulus of the velocity within the confines of the
layer. The coordinate x axis is taken to be a curved coordinate along
the surface of the body, where the y coordinate is taken to be the
length of the segment along the normal to the surface of the body.
We shall henceforth use two ideas in the investigation. The
first is that it is useful in certain cases to stratify the region of
1
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gas flow not only in the longitudinal direction, separating the laminar
sublayer, the turbulent layer, and the region of the external flow, but
also in the transverse direction, separating the following portions of
the layer: 1) the Reynolds layer, 2) the Oseen layer, and, 3) the
Prandtl layer. The second idea is to use the method of successive exami-
nation of the development of the phenomenon within the limits of the in-
dividual sections of the layer, with a transition from one section to
the other. This idea makes it possale to employ linearized equations.
1. Statement of the problem. We shall consider that the body
is stationary, and that the flow of gas has a velocity Uco at infinity,
directed parallel to the symmetgy axis of the body from left to right
(see Figure). We denote the angle between the tangents to the surface
of the shock wave ana the velocity vector of the external stream by p,
while the angle between the tangent to the surface of the body and the
same direction of the velocity vector we denote by 0. If we use the
known formulas for an oblique shock wave [4], derived under the assump-
tion that the viscosity and the heat conduction of the gas are not ta-
ken into account, we can obtain the m'ollowing equations:
sin sin (0 --8) cos cos ? 0)]
P A
1* A =c?1? ? sin cos(? 0)-1- cos?sin(?? 0)1
PA
CA I COS.1 + () sio PA p p sio p t
%PA \ PA
(1.1)
where the index fidenotes the values after the passage of the shock wave,
i.e., on the outer boundary of the intermediate layer considered by us,
the variable thickness of this layer being denoted by h. It follows from
Eq. (1.1) for the modulus of velocity VA that for values of angle p that
differ little from g/2, the maximum value of the velocity modulus Up, will
differ little from the value U?,,feceda4
U A "?'
r A
(1:2)
The ratio of the coefficient of viscosity to the density, /te../p
will not agree within the limits of the layer with the value ,40..v.900 ;
in the layer Tip, ee Tao we have .9,44 > Soo- The viscosity coefficient
increases with increasing temperature, i.e.lje.A e*.jevey.
Thus, it can be assumed that the order of the valuesimA/911.
is close to the order ofie,60/ (300 . In this case we obtain for
the Reynolds in the layer the following relation
2
e
?
?? ? ? ????? ??? ima....?????h????.??
P A L'Al
1?A =
Pc? 2_3P u.?3 tto
lie Po, P.t pA
(1.3)
If we consider the ratio 9ce/5),, small and of the same order
as e, and if we assume that.the Reynolda number of the exterpal stream
-?2nas an order of E72,- the Reynolds number fox .the flow of- gas inside
tee Intermediate layer considered by us will be of the coder el, and
means in turn that under these assumptions the layer between the
serface ehock wave and the blunted forward portion of the surface of
body can be considered as a Reynolds layer.
In the note/1/ we considered the case when ez:172 and the angle
,.ie almost T72, and did not take into account the variability of the
dzeeity withln the ,limits of the layer. The latter premise was also
t.7:k, starting point .in other investigations devoted to the same topic;
tees, in a recently published paper by Lee-Ting-i and Geiger/5/, the
jistanee between the surface of the shock wave to the critical point
the surface of the body is also determined by using the equations
of motion of the gas without taking into:account the var.:Ability of
T,:l6 density in the layer and without taking into account the viscosity
terms.
We propose that when the surface of the shock wave is closely
atacent to the surfaca of the blunted body, the influence of viscosity
seeeld be taken into account not only in the equations of metion of
the gas, but also in the derivation of the relations on the surface of
tee shock wave. The purpose of this article is indeed the derivation
tilt approximate equations of the Reynolds type, suitable in certain
,:aee$ le for theiflow of gas in a thin layer between a sUrface of.a- -
snock ,wave and, a blunted surface of the_bodyi and to Use in -
the aolution of these equations the .conditions an the peak, with allow-
ance for the viscosity and heat conduction of the gas.
2.2teuaticqzcd.R71112gALSSIllik!...119.11L01:11.2-812.211Y2Fe
We shall consider a section of the layer with abscissa x13:74
In this section we take the point B, at which the longitudinal velo-
city u has a maximum value U. The values of-the-other quantities at
the same point will be denoted by PB' -TB' apOss,,We_intro-
dutc the dimensionless variables and the dimensionless' Characteristics
gas flow in the layer as follows: ,
x lxi, y Ely], u
T = 7' B T
1.
1,132 p fJ12
?
?Pl.!
v = eti8 ,
P = PsPi. P= pn pt,
Pll
CP cpucti, ?
' Pt;
/WW2' /-
?en n
gepv Tu. (T ? ) ? pit
:LB
3
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;????1????????????????41.6.11.1?????
41.,????A.??????????1??? ?
:?? ; ?
of a
mails
when
?
114 shall consider the case of a plane-parallel steadrsstato flow
viscous and heat-conducting gas 'without taking into account the
forces. In this case the known equations of motion of the gas/6/.
using (2.4are expreased in the following form
ak, RH api a I NA
11 pi g- evi +
+ P? tel + k(1112-11)1
va) 4g2 71; (Pit +
gt4(14J2 +4[2+11(2 fa a?41
b(W1) a (Ps vi) = 0
?
sit aya
= (TO,
(2.2)
Rise plimi4017.1)? v1.41.(ci Ti)]. sit (311171+
re ill) np_.(x.M\1+ Lt. (0.,542
Pi I dzi 'Pr) aifs Oft /J kohlts /
? 1 ts te (21)1 4 [cos + (gals
Y
whore A is the heat equivalent of work. To obtain from Rqs. (2.2) the
well known equations of the .boundary layer, it is necessary to put
(21)
and assume that the numbers MB and PB are on the order of unity.
?
)16-1,
,
:
Putting then 0, vi obtain equations-for the flow in
the Prandt1 boundary layer. Since at 4F ,21, 0 the Reynolds number
increases 'to infinity, then the equations for the flow,in the boundary
layer will be the asymptotic equations of flaw of a viscous and heat
cboducting &I.e. On the other hand it we put
Rs 20 47, .461.64, Pgt.,1 .23)
and than decrease the parameter to 0, vi obtain from Iqs. (2.2) the
veil known Reynolds equations for the flow of gas in a layer, which
4
???
?
after transformation to dimensional variables assume the following
form
0 / =. 0 NEIA, as 0
? k.1.4. P Oy
p 40.8119T
a I. ka.
4 kt+ -40 1-AP-U.; , )k.lr (r);
(2.
Inasmuch as Eqs. (2.6) were also obtained by taking the limit
RB these equations can also be called asymptotic equations
for the flow of viscous and beat-conducting gas in a thin layer. The
difference between the Prandtl and Reynolds equations consist merely
in the order in which the Reynolds number goes to infinity, or the
order in which the ratio of the mean thickness of the layer to its
longitudinal extent diminishes.
We assume that to study the flow of gas in the vicinity of the
forward critical point on the body there are many grounes for using
Eqs. (2.6) rather than the Prandtl equations.
In reference/2/ we have shown that for the case of an incom-
pressible liquid it is easy to improve the solutions of Eqs. (2.6)
by using successive approximations. In individual cases of gas flow
in a layer, the solutions of Eqs. (2.6) can also be made more
accurate by representing the solutions of Eqs. (2.2) in the form of
series in powers of the small parameter ?.
3. Lipearizefi, equations for the flow of gas in the layer.
Eqs. (2.6) are in general nonlinear. In order to obtain
from Eqs. (2.2) linearized equations for the flow of --gas in a layer,
'we shall proceed with the following argument. For the nearest
vicinity of the considered point 1:1we assume that the dimensionless
variables are represented in the form
ul = 1 ? ate, rt = at'', = I +$!P, Pt = `?' (3.1)
T1 ara eT', t2 = 1 + eta', ).1 = I ft = EC1'
Inserting (3.1) into (2.2), using the assumptions (2.5) and
retaining only the terms with the lowest power of the
parameter 1, we obtain the following equations
C op' I Ai'
Y11-7-7 Oct cRn '
? .4- ?
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5
ioir
am. , ar?
sc").7,? P = = 0 0,.))
"wrif
?
s:??
I.
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.2glear
If in the resultant Nqs. (3.2) vs return to the initial
dimensional variables, i.e., we pat
a? 11 1
x1- r* '7
spa
2"-Ts Ps
7' ma , Ay .1;37 gTB epit ? 1)
a ?
(3.3)
then me get the simplestlinearised Ramada equations for the flow
of gas ia a laymr
?
Op 01/4 Op ale 0, t Po--;-7 = 0,
-g 2w Pa ! ay
D ? T
(3,4)
Oa the ?the:thud, i vs Put -
p-? = 1 +IP' ? ? (3.5)
V4 retain all the -remaining equations in (3.1) *td the tasusption (2.5),/
?
make the substitution in Xqs. (2.2), retain tha terms with the lowest
power of the parameter 5, and return in the resultant equations to
the initial dimensional vcriablosi wa *Lain a new &WU fora of
Unsuited Reydolla equations -
Op Otu dp
= 141 aTs ? 7; 7 v
+" iiY 7; 47; ? di) ?? 7Yr =
Ou 0, Op ar
?
114 eliainate the density froathe continuity equation with the
aid of the equation of state
(3.0
ali'ar_.7 P
Ph a Pa..
Oa the other hand, if we use instead of asmumptions (2.5), the
assuiptions in (2.3) and (2.4), again make me of all the
equations in (3.l, and repeat all the preceding calculations, we can
obtaia thefollowing-Ossea-type equations
-A au- Pa ,v,
(3. 7 )
P li v R -a-s' "'t ? 17; i', t 1 i V t " T, i sci: 0, p MIC yi L
17 au 8P` - a!" - ?Op Os'- 0,
gPe UStCPB+TBC;c01018L gm Al ID -:?:
6
If however we use aasumptions the_Oseen-type equations
arayepresanted in the form '
da lipPliv 117,7 ' 7;17 Fit,
7.c-F+u ()sp???(57- 17'"
Lie? , I op I dr,
dc
gPii B [e ps 11 it; Lip , . asT
d7' 7.; -- ? B*71-r- -1- Ali
a8)
The linearized equations (3.7) were used extensively in the
monograph of Targ/7/ to solve problems in the development of flow of
an incompressible liquid in tubes and dIffuscrs. The results of the
calculations were in satisfactory agreement with the results of the
experiments, not only with respect to the length of the initial portion,
but also in respect to the development' of the profile of the velocity
distribution over the sections. For this reason one can assume that
the foregoing linearized equations (3.6) or (3.8) can be used to solve
certain types of problems in the flow of gas in a layer.
1
???? .....
, Use
4. Dynamic conditions on the surface of the ebock wave. The
dynamic conditions that relate the characteristics of Ass 'motion with
the velOcity of discontinuity propagation, with allowance' for viscosity,
were carried out by Duhame and considered in detail in the work by N.
Ie Kochin/8/. These conditions can be written in symbolic form as
follows
pe'lvi-;
r
+.1Pffiv?VI ? pirley'r1+ ?111),,./.1
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OIL
1901
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???74
41.
T.Itatiette4
wherel* is the speed of propagation of the woive frontlithe velocity '
vector of gas motion, andln is the stress sedt vector on an area adja-
cent to the surface of the wave; the veator normal to the area is
directed towards the outer flow (ose.figure), In the case under con-
sideration by us, of a plane-walla flow of gess these conditions
(4.1), after certain transformations, can be represented as
? m
? P U002 Sint + (ron)A
PA
o.,
eA U.,[eoe ces (IF? I) ? sin p sin (1 0)
PA
? (4.2)
en (5 ? 8). (lim /A ?
VA "" [CCP". sin (ft ? 0) ? sin p coo (A._?sin (A ?*I ("ROA
- PA 78:17:
1/46 A (I I ? U4') + c ."" 44,74+ A
a
+A
PA
(4Cm)A1-4(11,E) A? A Oses),A in A cos (is? -f-rA sin al ?
8
PA /
0)1-0 (4.4)
The projections of the vector of the deviator stress an the
normal,t'nn, and on the tangent, 'ens, will be represented in the
following form
2 du
"317 I + 2IA ale(_e) cos= (0 _
? ay au.
? 7(17
Tn. = (47?a; ? -T-tx) sin 2 (II ? 0) + IL (Ph + -a-4) cos 2(? 0)
(4.5).
If in the right halves of Eq. (4.5) we go to dimensionless
quantities, using (2.1), and atm retain the terms of highest order,
we can Obtain the following approximate expressions for the projections
of the deviator stress
Cu. 2 (ft e),
tme?P--- cos 2 (13 ?9)
(4.(i)
Thus, the relations on the surface of the shock wave, lith
allowance for viscosity and heat conduction, are represented by Eqs.
(4.2), (4.3), (4.4), and (4.6).
5. General expressions for the characteristics of gas flow in
the intermediate layer. The simplest linearized equations (3.4) were
used in our article/1/ under certain supplementary assumptions. We
shall now consider the use of the system of Eqs. (3.6) with partial
allowance for tho variability of the density.
If we retain the previous notation, but consider not a plane-
parallel flow but a gas flow with a symmetry axis, then the linearized
equations, with partial allowance for the veriability of the density
and with approximate replacement of r by the lengths of the arc x,
mill be represented in the following farm
ap alu a a
to.a .5-40 + 70-r- [x unfa?ulinj
a P A ?IT A p , A T , p = = , ? mit ? ?,?
?
ay ' ay' pn J B pis
(5.0.
The boundary conditions on the surface of the body and on the
surface of the shock wave mill have the form
9
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.1Y
,r? tat ? ? ???:?.a.n ? ? ?aaff
U =0,
14 5-- 141,
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v=0, T 7' ? npm y
r r,j, T T, npt y h
(5.2).
?
The solutions of lqs. (5.1) sUbject to boundary conditions (5.2)
viii be given by the equations ?
XX=
? ?PA (ys
21411 Os
r
LX (ig s
0
?
NA
hY)-}- -7-4- y
tot, _tin T Idyl
Pm rit
T=.7:2?.-1-(TA*?Te)-1-
?
(5.3)
If in the second equation of,(5.3) the-upper liait 7 is .replaced
by h, the operation of differentiation with respect to the variable
x: is taken outside the integral sign, and the first and third equations
of (5.3) are sat then used together with the boundary condition (5.2)
for v, we obtain an equation for the pressure '
It I is dpAN go i h I digA '
1441ta -2- -r.
'h \ L. 14TA \
TA)(z- + -7) n k dx -277.1;
Ikeda the first Sq. (5.3) we get
.? ?
?
(Ou\ h dPA j_idA
k as ZIAm dist
(5.4)
(5.5)
Inserting (5.5) into (4.6) and then into (4.2), (4.3), and (4.5)
we obtain the following expressions for the Characteristics of gas ?
flow on the vary surface of the shock wave -
PA Ps lifts SW" ? sin 2 (11 ? 9)14}441 FIBNA}.(5.15).
?
VA
+ co_201 ? II) coma (a ? 6)
L 10077407
SO lift [cos fi c*(_ I)+ ttt sin sin (p ?11)) ?
PA -
2 ?
lo
4 1.
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dPA ? $) cos 2 (iS
dx dais
?
????????,,,
1
?
VA = Um[ces 13 si ?0) ? -sin p cos tp ?
1B
!1I
coh A
dPA sin (a ? 0) cos 2 (5 ? 6)
sin a 2peo Ue, dr sin a
crA T A-- ercoT co + PcoU)0,?Asin a A h 6 =A IT (11C: U A2)
4_ Pal 112_3 MA 4_ ri dPA sin (3 0)
Pco P.t PA \ A 2111J thr ?
2(ti ? e)[ [uA cos ? 0) rA ? 0)I}
Eqs. (5.4) and (5.6) must be used in conjunction with the
geometric relation
dA t--- g(f) ?
dx
We chose as the characteristic velocii.y11Din, se ction 2 the
maximum value nor the longitulinal velocity u in the section Vith
abscissa xBz. Z. Since we have
dpA MA
7-ix- (2Y I"
be
from the ordinate of the point BAdetermined by the equality
we get
Pft "A
Ym ? 2
lidpA,dx
Inserting (5.8) into the first and last equations of (5.3)
U m mit
dpA ( A1JVMA 1
21AB dX 2 A dpA dr
qft U A
T B re + (TA 4 ) [ 2 ht dpA dx I
(5.0)
If we use the approximate equation of state in this case we
obtain the following expression for the density
TB
Ps PA 2Tii TA
a
.1.
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?
Since the initial equations (5.1) ate morrect for a section
with abscissa xR 1, then the relation (5.4) and relations (5.6),
(5.9), and (5.10) are correot, generally speaking, for .4a, emelt
vicinity of the intermediate layer considered by us near the fixed
section. As 004 404011 away from, the given section towards the broadenr.
ing of the layer, the velocity 1111 increases and the Reynolds number
Ifitirma ?suits (W)
,
(UA)X4 4 0
WAN.* = Uco = kif
rA
. , ? '
(PAA-2= (1 ? k)pcd; 022
(6.5)
(tvA Tit ? Tee 44.A tio:+19 .04) zrz .4 {:-5-'1 (1 k) + 7,14- U cot (1 ? k)s}
Differentiating the second Eq. (5.6) and using the foregoing
will also increase; as one approachea the symietry axis, this Reynolds
nudes? (5.11) will decrease. Consequently, the vicinity of the layer
for which the faregoing relations are valid can be extended with
towards removal away from
equations for the
(d-4-j
X40
limiting transitions, we get
t af? + k ? de) ?
NAV,. att 10. .11).?K;
d12)?}
ram3
(6.6)
lesser error towards the symmetry axis.than
1
this axis.
6. Limiting relations for the symmetry axis. Lt us assume
that the relations established in section 5 ire .correct also for those
sections of the intermediate lver considered by us 'which are
sufficiently close to the symmetry axis of the blunted body. Subject
to . . this assumption is perform-anthem relations the transition
to the limit, decreasing the abscissa x to 0, increasing the Ingle
tor/1, and petting 141772. We then have
cosp7-,0, - 0) - 0
coe ? 1, sin 2(?e)-,O, coe2(p?O)-?1
Using the LIRospital rule we get ?
1"1-1
11140118.
I. z _IA
??? ds .
li
P.446
(6.2)
Mance on the gymmetry axis the pressure should have a maximum
wars, we get
Pros rq. (.5.7) ie get
(62)
111411". (844
Using (6.1) and (16.3)ve obtain from (5.6) the following Uniting
relations
12
If we start with the first Eq. (5.3), we find that the
longitudinal velocity along the entire symmetry axis equals 0, and
consequently me can put
Vs=c (1
From the Litinspital rule 'we get
ii113
X )X +0
du A
'
'PAI _dtPA
bin
dx J:c .0 dr'
(6.7)
(6.8)
Melting the limiting transition in Eq. (5.4) and using (6.5))
(6.3), (6.4)) (6.8) and (6.7) is get ,
, ? 1473 duA
kl'e?.:--.16 dr; ? h. 7;1
xmo
(6.9)
If We differentiate the first equation of (5.6) twice and then
make the transition to thelimit;'we,obtain
(6.10)
?;
k410-) ?12(1 - k)Pc411?31(dY + 4 dd-x.3- dde?, 'F.' yr? 7-du: Dx=0"
xseo
Inserting (6.9) into (6.6) and (6.10), me obtain the following
two relations
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?
(20(4+
4 vw:1-470)=
vbere
4.0.4444.11144?4.1,
----tr?vt.t.Ti-vt!rmrettt
ete k
+i L
ma. ? esTu;i5L0
cr
44ts?
tttt(g-
0 dig ?27.?fiall
d2p/ds2 from (6.11) we get
COIL
eV+ 2.14+ c 0
a as + (1k) k(" ?I )
1*1
(6,12)
4
bia--114k4-1-44-(1-2k)kl,1 (6.13)
p.47:7rsh "1-: 7 4? ii`Ms}
Solving Eq. (6.12) and selecting the sign in front of the Taus
root in this saution on the basis of the coaditiomthat the Ingle must
decrease with increasing x, we get
kV,? ?44 +
(6.14)
Thus, if we assume fim,,140 en3(00/dx known, specify pre-
liminary values of h*, k, endif.B OA ths basis O $ome other considera-
tions, we can determine the valle of (d0/0)0 from Eq. (6.14), and
will therefore have in the nearest vicinity oe the eymmatry Axis
= ( diqs1 )41x
From the foregoing data and from Ihe vape of (4/6)0 as
Obtained from (6.14) we can determine Oft piox-)0 ttix:Itthe first
Eq. (6.I1), and determine the value of (daii/dx)0 frail (6.6). We
then have for the vicinity of the symmetry. axis
40.*
P ma (I 4e -1- _2/ uA k.--z x
? (6.16)
-
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Next, going to the section close to the axis and determining the
angle?) for this section from (6.15), we can repeat the entire
argument and derive formulas analogous to (6.14), (6.15), and (6.16).
The last equation of (6.5) contains the temperature To on the wall,
which cannot be considered assumed. In the first approximation this
temperature can be assumed equal to the temperature TA. If one
assumes that the coefficient of heat capacity cv. is represented by
a definite dependence on T, it is possible to determine from the last
equation of (6.5) the temperature TA in terms of Top, pow, Utp, fool
and k. In this case it is possible to determine from the temperature
TA the viscosity coefficient on the axis and putp-B=IAA, in
Eqs. (6.13). Then, if Eqs. (6.13) are used, it is
enough to specify the tentative value of the thickness of the layer
h* on the symmetry axis.
Received 17 February 1958.
BIBLIOGRAPHY
1. Slezkin N. A. Concerning the Problem of Determining the Distribution
of Pressure on the Blunting Area of a Shell, pa SSSR (Reports of the
Academy of Sciences USSR), vol 54, no 7, pp 583-585, 1946.
2. Slezkin N. A, Concerning the Problem of Refining the Solution of
the Reynolds Equation, DAN SSSR, vol 54, no 2, pp 121-124, 1946.
3. Reynolds, 0. Eydrodynemic Theory of Lubrication and its Application
to the Tower Xrgpc.,111ante. Coll. (ldrodinamicheakoya teoriya
saki (Hydrodynamic Taeory of Lurbication), GTTI, pp 249-360,
1934.
4. boytsyanskiy L. G. Mekhanika zhidkosti i gaze (Mechanics of Liquids
and Gases) AITTL, p 326 (1957).
8. Li Ting-i, Geiger R. Crial Point of a Blunt-Nose Body in a
Rypersonic Stream. Coll. Translatione "Mekhanika*( Kechanics)
5, pp 33-48 (1957).
6. Sohlichting G. Teoriya pogranichnovo eloya (Boundary Layer Theory)
Buss. Transl. from German, IL, p 254, 1956.
7. Targ S. K. Osnovnyye zadachi toorii laminarnykh techeniy (Princi-
pal problems of the Theory of Laminar now), G1TTL, 1951.
Declassified in Part - Sanitized Co .y Ap roved for Release ? 50-Yr 2014/03/14: CIA-RDP81-01043R004200140002-4
4. , IS. .? ? ????? ??? ? A 0.4...4.4 4... C.44 .1. .4 - ? .144 4..44 ? 4. 14 an., oft. et ..1.43.44?44.14.4.4,4.4
4
8. Eechin N. Ye. Sobraniya sochinenly (Collected. Works), vol II,
pp 6--42, Isd. AN SSSR, 1949.
,
.16
? .? .? ... ? 44, 444 ? 414.4.4464.4*? 4.114.114.411
? ? ? ?
Amioximation Method of Calculatinr Shock Waves ?
and*- Waeir Intera ions
Isvestiya Akr.zdeinii Nauk SSR .0t- .
deleniye llbkh.nicheskikh Nauk .-
Mekhexiika i-Masailostrontatzt, 'Mews
of the Academy of Sciences USSR, Dep-
artment. ofTectuiical Sciences, Mechanics
and Machine Btd.lding], No 2, Mar-Apr 1959,
Moscow, pages 13-18. ??
1. Description of the method. The problem of propagation of
..lint,ona stationary Shock -Wave has at ;resent not been solved in general
form even for unidimensional plane flow. The system of three quasi-
lineer,,,fir.st-order. partial differential equations
?
:G. M. Iyakhov
N. I. Payakova
Ou au Op ap Op au
71- + = ' ? ? + p = 0,
r p ax Or ax
,
Whioh describes the non.-stationary-shookave, should be solved Subject
to the boundary conditions on the front of the wave, on the line
that is also sought and'must itself be determined from the considered
system of equations. This circumstance complicates the Rik solution of
the problem considerably.
Os as
Tit + = 0 (1.1)
The method proposed is based on the fact that the (=von that
expresses the law of compressibility of the medium, p= p ie
replaced by a broken line with segments of the type
(1.2)
where A and B are constant- within the limits of each segment of the
broken line.
'
Such an approxAnatd.oa yes first used 'byChaplyginN in. a
consideretion of stzt;..iy?-,.1.4,;,3 flov of ir.:.3) rInd vas tail? used by
L'.I...Bed-pv/2/ and, iLtLas by .15.,P.
In dense media the lict losses can be neglected at pressures on
,
the Order. of 'tens or even kti.raireds g.f-ittmosplaerefil and therefore this
methOecan find vie u here; However; such.-an approximation is
possible a1s6 in an analySis of ,in the case when
the pressure on the front doea not exceed-.. Or-.3 kg/cm2. The
liugoniot adiabat, which gives the connection between the .pressure p
and the'specific ioliime V *on the front of-?the -wave'
=-1? (lc ?.1)p4-(k*1.)pe
? -.4,-7 ft 1 (kzt 1)p +. (k???)p.
differs in this ease tittle ix.O.ni.:Cie.-PoiSson'
P ?Pe 0
?
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(1.3)
(1.4)
?
MEW,
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For illustration, we give the.ValUes of the volume V, calculated
in accordance with (1.3) and (1.4), for certain value of P/P0
rift-a 1.0
V/Ve=1
V/VioNg
I.
0.750
0.741
2.0
0415
0.600
2.6.
0.531
0.52&.
accordihi to
according to
(1.1
(1.4
We write down the basic equations for one-dimensional plane case
in the Lagrange system of coordinates
Ja
11'
ow eV A
-- "
(1.5)
Here h is the mass of the substance between the initial and the
current sections
ffitl PdZi
0
Let the equation of state of the medium be
Then the system of equations becomes
Oa . ar A OA I Op A
From this we readily obtain the well known wave equation
asp 1/_!e.
ZS.
the solution of which has the form
+ rso + AO,
,6)
(1 .7)
t .
x 71- Li? 1(h-At)-- F2(11-+ Atli (1 .13)
Here F1 and F, are arbitrary functions, which should be deter-
mined from the boundary conditions, A is the velocity of propagation
of small disturbances in coordinates h,t. The quantity A corresponds
to the acoustic impedance of the medium Ic for a given section of
the approximation of the isentropic curve.
2. Propagation of shook waves. We consider the Propagation of
a plane shock wave in a medium, the equation of state of which is
3iven in the form pap (V) While the pressure, as a function of time,
is defined at a certain section of this medium, which we
shall consider the kitial section. The boundary conditions are
18
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opy Approvedor Release
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specified in the initial section and on the front of the wave. The
lam of motion of the front is not known beforehand and must itself be .
determined from the system of the principal equations.
The boundary. conditions on the front ofthe_shock wave are
of the form
'p1)- p
,
?
Da. (2.1)
e P go- are the parameters of the medium ,in front of the-
wave front,e p, 3, D, and n- front are, the parameters on the ont,of
the wave.
'r?''S.
Ellmtnating u. from these equations Ve.ge.t2
Pe ?=7- Pa21)2.(V0 V)
If the curve pp (V) is approximeW,W anetraicht
p .42 (V -V0), thenf0D-ztA, i.e., the velocity of the front of
the gave coincides with the speed of propagation of the weak dis-
continuities. In this case all' the states behind the front of the
wave move with equal, velocity, equal to the velocity of the front.
The wave will proceed Without damping and without a change ia its
front. Let us consider the case when the curve p p (V) is approxi-
mated by a broken line (Fig. 1). On the section closest to the front
we have
Akar,e(
?
-,?
P --.4?211 +
of the front we have
-4.2V-1- Be
-
- 1"
The velocity of the front in Lagrangian coordinates is
+ P77:71P Fa
0
It ie,rebvious that
Api Pn- > > At = 17-r=a-v
Y40-1 ?- -
19 ,
(2.3)
(2.4)
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I.
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?
.44
where 41 corresponds to the speed of propagetion of the states on
the first section, and An corresponis to the last section of the
approximation. It follows froa this that the states behind the front
of thensave propagates more rapidly than the front itself, and the front
ital.? moves more rapidly than the yeak disturbances in front of it.
Thus, when the curve pep (v) is replaced by a broken lino with several
segments, the shock van -changes its shape as it propagates. The
magnitude of the maximum pressure will decrease. By reducing the
sizes of the segments, it is possible to obtain any degree of accuracy
in the determination of the 'wave parameters.
It follows from (2.1) and (2.4) that
p ps 9.(V ? V ?), u (V ? ? Vs)
Taking (2.2) and (2.3) into account, we get
oaks* atnie?
P.-- Po = Ass_ k.s.4.P= h.1
1lsoB,-4?AD (2.)
The flow between the section bax0 and the front of the wave is
determined by expressions (1.8). Henne we have on.the front of the
wave
2F1 lor p Asu, 2Fs p Asx
In accordance with (2.6) we have on the front of the wave
Here
2F, ?p.-$? 2Po.si pa? xi"
."
(2.7)
gs? 474.046,
If the approximation segments ars sufficiently small, then,
as calculations show, the function? can be replaced on each of the
20
44.4-141,1!
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?
? Ar? ? +4 ? y ? 11 p.1, .?????.:4.??t*.l...6!'","! ?
c.
? ? ?
011.????????
? ,
segments by a etraight line, while j0, can be aseumed constant. Such
a linearization of the boundary conditions makes it possible to obtain'
a solution it an explicit and readily visualized form.
If q2 (hf)lacorat, then F2 (h1FAnt)vg.const on the line of the
front, and consequently, in the entire region 1 (Fig. 2). Ii%om the
condition at the section het? we determine the function Fi Oa -Ant).
Let, for example, the change in the pressUre be specified in the
section hme0 in the following form
Pg'144.: it (e..a po+pss,
We then have in the section h=0
Ark+ Fa.104-1-bc,
Hence in the entire region 1
pis ft+ Fiore? -f-tqh
. Alx ?
x va (F% ?Fx)iot 7:: LA. 24?
(21)
The solution obtained viii be correct in region 1 between the
section hs0, the front of the veva, in the otraight line
lisinAm(t?r?..4), 16_1 zr2t2if71'
wherelaa is the instant of time when the pressura in section hat()
drops to a value corresponding to the lower limit of the pressure
on the n-th section or the .approximation.
Let us find the second boundary of the region 1 -- tbefront of
the shock wave. Since the function 11 is determined by (2.9), then,
inserting (2.9) into (2.7), we obtain a diffemntial eqpition for the
motion of the Tows front
where
? ?
f
264
le ka ks Ai. 2a po Aou. ? (2.12)
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23.
??? 0.? ??? ???
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Solving this equation and takihg into account the fact that
h =0 when tee0; we obtain the law of motion of the front of the shock
Wye
lit,.ARat-1-k -1-an--14kni-e,X-F4m.bli
[ n
(2.13)
Let nowp-e.e--plin the section h==0. The region 3 the
hit plane, where the pPeiUre corresponds to the (114th approxi-
mation segment, cannot make direct contact with region 1, since the
straight line an which peeconst has a different Slope
in the nem region than in X. Between the regions 1 and 3 there
should lie a region.2 of constant parameters, bounded by the charac-
teristics. For the region 3 the corresponding calculations yield
1 f
u= A71:31a ? 21:2 ;TT:10 ? An_11).1 (2.14)
A r
21-1 21)(1---t)+kn-2-1-an?i? (k n_q+a1,1)24a,1-114( Ilk (2.15)
Here h f and 'ears the values of h ancteat the point where
the front starts out in region 3.
We determine analogously the flow in the succeeding regions.
Thus, for the pressure specified at a given section of the
medium, we determine the law of motion of the front of the wave and
the flow behind the front.
3. Reflection of shock wave. Let us consider the reflection
of a plane shock wave from a rigid partition. In the case
of densasingle-component multi-component media, there is no doubt
that the approximate method considered above is applicable to the
solution of this problem, since the heat losses dueto reflection
are small. In air the problem can be solved by this method only at
smail-pressures. However, AS the front of the Wave reflected from
the partition Moves into the incident wave, the pressure on the
front of the reflected wave decreases, and with it the entropy jump.
To the contrary, the entropy of the pssticles in the incident wave
increases with the distance from the partition. Therefore there
occurs during the reflection an equalization of the entropy at
various particles of the medium.
On the front of the reflected shock wave we have
1(h- u1) p2 (1) -- ut), pe? pi (D- u,)(u2
22
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?
'11
?there the index 1 corresponds to the incident Wave and the index 2
to the reflected one.
r Let the states behind Vla front of the reflected and incident
waves lie in different sections of the approximation
ANI -I-A, Ps se ?As2i 1-4-
Assuming u2:10, me obtain an expression for the pressure p
e
he tftX
in the reflected wave, acting on the parti
instant of reflection,
Adis rih
? ' 1.7t? - -
-? Lit
TA-- A&&?Pitill4 Liz.
tion during t
PiUk74A ?
I
(: 44) 11
s AO
?
,(3.2)
In the case of air this p formula is practically the same as
the Izmaylov formula. Taking into account the'connection between the
pressure in the incident and reflected waves at Al A2, we'find
that the plus sign should be taken here.
Let us denote the line of the front of the reflected wave by
7faghf (t). The solution in the region of the reflected wave has the
form
Fah ? +(Pi Aluz), FLO + A. ? (PL? Alm.)
The arbitrary functions must be found frOm the boundary condi-
tions. The conditions on the front yield
(3.3)
(P, s2CFALs14--.. Pt
4111:11Bs,
E4:-.--1. P
2' Vig :+1141;t:s at;
Here
A
(3.4)
Unlike-the incident wave, the variables in the expressions
Fi and F2 are not only the functions ?Pi and r2, but also ul and
wnich comptcates the solution.
If the reflection is from a stationary partition, the second
2,
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MA Sir Os
???? ? .
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? ? ? .=? ? ? ? ?
ri ? " Fs.Ve Ast)
where h* is the coordinate of the partition.. -
If the reflection is from.a moving partition
ma tix= it,
3q,V, )614 -- .4600 am A fOth (k? Aso -- (Ds 04 .4- Am
where the Index: 3 corresponds to the parameters of the mediumibehind
the partition, m is the mass of the partition, and uis its velocity.
(3.5)
Let um ',midair thirefleption are non-stationary shook wave
fibmi'a silt:Emery pertition. Without limiting the generality, 'we can
assume that the incident 'wave riaisfies the partition on that section
of the path, where it is stationary (Fig. 2). In region 4 all the
parameters are. constant. Then they will also be constant in region 5.
The Velocity of ,the particles in region 5, is %et?, and the pressure
is determined by (3.2). The yelaCity-ot.the frZat of the reflected
wave is
(3.7)
The' boundary Of,region,5,is thecharecteristic a .
h Air so coca
, Wars
which is drawn from the point appiatim the line of the front of the
reflected wave meets the line that bounds raglan :4. ,The characteristic
b limits the region 7 on the side of region 8 and intersects the front
of the reflected wave. The equation of this characteristic is
- h + Astra coast
' '
in region 5.and in region 7 'we haveir; (h4420.1x 412/2
The boundary condition on the front of the 'wave is
MI
2(k-+ Ast)" PA? As*Aim ?Aeu?s ; (3.8)
_
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Considering that in the incident.Mave the pressure and velocity
of the particles are determined by (2.14),. me obtain the differential
equation for the motion of the front
Here
riq
A '011 ars.? Poi) ( 4- 41) - A -,t+
P"2f,. Pi it
AL-- Ai AO-
.ge *3(4,
AA 51A2. Si
1)
The solution of the ,equation is
of ma," t+ cfilist (3.10)
Thus, the line of the front of the reflected wave represents a
second-order curve. The function 511-(h -A2t) in region 7 is determined
from the relation
?AA ps.-+ Aos
ag* pi + A# 4-1 1 1 23 21 A391.
ih
? ? (Ast? + ?AO A
which is satisfied on the now known line of the front of the reflected
wave.
The limit of region 7 is the characteristic
h const
which is drawn from the partition and intersects the line of
the wave front. In :egion 8 between this characteristic and the parti-
tion, the function(h -A2t) will be the same as in region 7. The
function .ti
(h4.42t)lis determined from the condition that wart) on
the parti on. The solution in the succeeding regions (Fig. 2) is
obtained in a similar manner, with simultaneous determination of the
line of the front.
We assume that a certain instant of time the pressure on the
partition in the reflect wave has dropped to a value p*, starting
with which it is necessary to go to the second section of the approxi-
mation of the isentropic curve.
On the line p (h,t)A-p* (dotted line), which is 'not a charac-
25
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teristic, all the parameters are khown, and therefore the flow behinl.
this line: in tha regions bounded by tha corresponding characteristics,
is fully determined. If, as in the examples conaidered above, the
functions Faland F, are linear, then the line plme will be broken, and
will consist of straight-line segments (shown 'dotted in Fig. 2). The
flow in regions 10 and 11 is dstarminid starting with the conditions on
the lines pmcp0. The flow in region 12 is determined starting with the
conditions on the characteristic and on the partition, while in region
14 it is determined frowthe conditions on the characteristic and on
the front, in region 13 it is determined from the conditions on the
two characteristics. The further flow is built up in an
analogous manner.
The authors are grateful to L.I. Sedov and LP. Stanyukovich
for attention and interest in this investigation.
1.
Received 9 Oeptembar 1958.
BIBLIOGRAPHY
.chaplpgin S. A. 0 genovykh struyakh (On Gas Jets), Oostekhisdat,
1949.
2. Sedov L. I. Ploikire sadaohi gidrodinaniki I aerodinamiki (Plane
Problems in Hydrodynamics sad AerOdynamies), Goetekhisdat,
1950.
3. Stanyukovioh I. P. Hew Approximate Bethod of Integrating Certain
Hyperbolic Notations, MUM, vol =II, no 6, 1953.
lig. 1.- Scheme for approximating
the curve p *30
26
11
11
lig. 2. Diagram showing the
regions of the various
solutions.
"4!
13
Deceleration of a Supersonic Flow in "and Tunnel Diffusers
Izvestiya Akademii Hauk SSR Ot-
deleniye Tekhnicheskikh Nauk,
Mekhanika I il.ashinostroyeniye, [News
of the Academy of Sciences USSR, Del>.
artment of Technical Sciences, Mechanics
and Machine Building), No 2: Mar-Apr 1959,
Moscow,pages 19-24.
We report in this paper the results of an investigation of the
procs o deceleration of supersonic flow in the converging portion
of diffuser channels of wind tunnels. Criteria are determined Dor the
maximum possible deceleration of the flow to the narrow section of the
diffuser, which will be called henceforth the throat. The effect of
the Reynolds number on the characteristics of the diffuser is investi-
gated. Based on the experimental data, an approximate procedure is
propoled for the calculation, making it possible to determine the
effectiveness of a diffuser channel of a given geometry.
N. N. Shirokov
An experimental verification of the computation procedure is
made for different Mach numbers.
A characteristic feature of the published results of research
devoted to the problem of deceleration of supersonic flow in wind-
tunnel diffusers is the absence of any method whatever for toe pre-
liminary calculation of the coefficient of pressure restoration,
with the exception of flow calculation based on an ideal liquid, the
results of which, as a rule, are quite far from the experimental data
obtained. This is evidence that the problem of preliminary calculation
of the pressure recovery coefficient depends essentially on a knowledge
of the laws of the influence of viscosity on the deceleration process.
The process of deceleration of supersonic flow in a diffuser
can be broken up into two stages -- the reduction in the
supersonic speed in front of the blocking shockin toe converging
portion of the channel, and the deceleration in the blocking jump
itself and in the channel behind it. In investigations of the adjust-
able diffusers of wind tunnels there is always a clearly pronounced
maximum in the relation or.gf En, whereeris the recovery coefficient
of total pressure, and F is the relative area of the diffuser throat,
i.e., beyond a certain valve of the throat area, further de-
celeration of the flow in the converging channel does not lead to
an increase in 0; but, to the contrary, it leads to a sharp decrease
in the pressure recovery *coefficient. Various authors have ex-
plained the character of the curve 6.f fF) by the fact that as the
velocity in the throat is decreased, the losses in the blocking snook
decrease more slowly than the increase in loss in the flow behind the
27
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blocking shock. However, experimental data do not confirm this
hypothesis. Among the problems still unsolved concerning the
deceleration process are two principal ones -- what determines the
maximum possible deceleration of flaw in the converging channel, i.e.,
and what determines the maximum in the relation dr-ef (F), i.e.,
tne optimum geometry of the diffuser. The present paper is devoted
to a clarification of these problems.
Description of the experimental setup. The investigations were
carried out in experimental setup (Fig. 1) consisting of receiver 1,
in which a regulating valve 2 was used to maintain a given pressure,
which is registered with a standard manometer 3. The air for the
experiment was taken from a tank, of high pressure air flasks. Conn-
ected to the.; receiver were interchangeable flat nozzles 4. The inves-
tigated diffuser channel were connected to the nozzles. These consisted
of stationary sidewalls 5 and movable eyelids 6, the number
and shapes of -which could be varied from experiment to experiment over
a wide range. The cross sections of the diffuser channel were changed
by means of special screws 7 and the accuracy of the displacement of
the eyelids was .40.1 mm on eachside, and was registered with indica-
tors 8. A non-adjustable subsonic diffuser 9 with a throttle 10 on
its outlet was than attached to the movable eyelids. During the time
of the experiments, a measurement was made of the pressure in the re-
ceiver, using manometer 3, while the distribution of the static pressure
along the symmetry axis of the latter malls and of the moving eyelids
was measured with mercury differential manometers 11. The fields were
teem= traversed by fittings for total and static pressure in two
mutually-perpendicular directions, 12, at fixed time intervals along
the length of the diffuser.
It was possible to observe and photograph the flow through avt.
LAB-451 instrument and to photograph the process with motion picture
camera SKS-1. The principal investigations were carried out at flow
velocities corresponding totette3.0.
Results of the experiment and their analysis. Fig. 2 shows the
distribution of the static pressure along the axis of the sidevall of
the diffuser at a minimum value of throat area for a given channel
geometry, along 'with the . . shadow photograph. corresponding to this
distribution. Judging from the shadow pattern, the reflection of the
jumps from the walls occurs even in the subcritical region, where the
influence of the viscosity does not go beyond the of the bound-
ary limit.* The same figures shows the pressure distribution cal-
culated for a flow'of an ideal liquid in the sage channel, with a
correction introduced for the thickness of the volume displacement.
* The results of the investigation of the interaction between
28
0
Let s estimate the calculated flow pattern from the point of
view of the critical ratio of the pressure in the shock waves. The
value. of the relative pressure in the shock ,waves when the latter are
reflected from the walls and when they.intersect the sideualls are
shaft in Fig. 3, which also shows the curve of the critical pressure
ratio. It is seen that in the investigated channel the pressure ratio
in the shocks at a minimum throat area does net reach critical value
and consequently the curve of the critical pressure ratios for this
case is not a criterion capable of determining the maximum possible
braking of the flow.
Let us consider the change of the velocity field in the field
of the diffuser as the throat area is reduced (Fig. 4), and the change
in the velocity field along the length of the diffuser for a minimum
throat (Fig. 5). It fellows from the examination that as the throat
is reduced the velocity profile between the stationary lateral rails
becomes substantially less filled, and approaches a detachment
profile. The velocity profile between the movable eyelids is
also deformed, but much less. The deformation of the profiles occurs
principally on the finite portion of the converging part of the diffuser.
After the swing of the stream in the throat, there occurs in the di-
verging channel of the diffuser a filling of the velocity profile on
beth walls. It follows therefore that the weakest place in the channel,
from the point of view of 717 a closeness to detachment, is the
velocity profile on the lateral wall in the throat of the diffuser.
It was shown in reference/1/ that upon detachment of a stream moving
with a positive pressure gradient, the criterion that characterizes the
state of the boundary layer, namely
i= -
where pf e of the x is the first der ativ
14.2 pressure
respect to
the length
thexa in a given section of the boundary layer, z is the characteristic
dimension of the.boundary layer, ctis the density, and le.is the velo-
city of flow on the limit of the boundary layer, depends
little on the Mach number (La the investigated range) and, when calcu-
lated in accordanee 'with the aerodynamic volume displacement 6*, has
an approximate value of 0.014 or 0.015.
A calculation of the values ofSin our experiments shows
that as the throat area is decreased, the value oftincreases and
reaches a maximum value at Finiq, approachine in absolute
the boundary layer and the shocks and the determination of the curve
of critical pressure ratio in shock waves were reported in a paper by
G. I. Petrov at the session of the Department of Technical Sciences
of the Academy of Sciences USSR in June 1958.
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29
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magnitude the critical valeta (Fig. 6). In our experiments, after
measuring the velocity field, it is impossible to fix directly the '
detachment of the stream, i.e., to obtain*,
t since the detachment
of the stream in the throat MOO= disturbs the supersonic flow in the
converging portion of the diffuser channel and partially in the nozzle.
To confirm the foregoing premise concerning the detachment from
the lateral wails, we exit took a high speed motion picture of the
shadow pattern of the flow in the region of the diffuser throat. The
pictures were taken at 4,000 frames per second. Fig. 7 shows frames
of the motion picture film, fixing the flew at the instant of detach-
ment.
Photograph 1 shows the normal floe pattern prior to the detachment
a supersonic flow with oblique shock waves in the converging portion,
turning of the flow in the throat of the diffuser, and the acceleration
of the flow in the diverging portion of the diffuser channel (flow from
left to right). Photographs 3 to 6 show in the throat of the diffuser
an the sidewall the formation of the detachment zone (dark spcf ) which
moves against the flow (photos 9 -- 11, 13, 16).
Photographs 11, 13, and 16 show clearly that the detachment zone
follows the produced shock wave, changing its shape as it moves towards
the converging channel. During the motion of this shock, supersonic flow
Is retained in the throat of the diffuser and in the diverging portion
of the channel. Photographs 18 and 20 show the formation and the motion
of the second shock wave, which differs in shape from the first one,
since the velocities in the converging channel have been reduced after
the passage of the first shock wave. This is followed by the formation
of new compression waves, photographs 23, 24, and 27 and finally, the
velocity of sound is established in the throat, photograph 32.
All this complicated system of shock waves stops moving after
it reaches the corresponding section in the nozzle, and a subsonic
flow is established in the converging portion, photograph 37.
It should be noted that during the entire time of the separatbn
process, approximately 0.01 seconds, supersonic flow is retained'in
the diverging channel, with a corresponding blackleg shock.
The results of this experiment give direct confirmation of the
fact that the maximum possible retardation of the XIV= flow in the
converging channel is determined by on the lateral wall.
Measurement of at Flan was cartied out in additien in channels
with different lengths of converging portion and with different bound-
ary layers at the inlet to the diffuser. The results of the measure-
3.?
ePot.
e shown in Fig. 8 and confirm the aseumption of weak dependence of
* on the Mhoh number (in the investigated range).
The usual procedure of investigation of diffuser channels
provides for a reduction in the pressure in the receiver after the
starting of the tube and the establishment of the necessary through
sections in the diffuser. Thus, the determination of the relationships
Orr-f (P) occurs at variable R numbers. To eliminate the influence of the
variation in the R number, the characteristic 6=f (P) was plotted at
ft constant pressure in the receiver, and a throttle at the outlet from
the diffuser was used for the determination of tr. It vas found in
these investigations that the descending branches in the char-
ecteristicG6f (F) are absent (Fig. 9). What is the same, at different
pressures po in the receiver for a channel of given geometry.
Approximate calculation of the characteristic 6f (r). Based
.on the data of the preceding sections, va can calculate approximately
the effectiveness of the diffuser channel with an accuracy sufficient
for practical purposes. The calculation is broken up into two stages:
the first is the determination of the limiting curve Frnineef (p0),
which depends on the Mich number of the nozzle and on the thickness
of the boundary layer, on the shape and dimensions of the converging
channel. The second stage is the determination of Fmin corresponding
to the maximum pressure recovery at a given geometry of diverging
channel.
In the determination of it is necessary to know the change
pa the throat of 141 P' and .01; this can be obtained by calculating
the flow in the converging channel with allowance for the boundary
layer.
To calculate gle it is possible to use the wel: known procedures
of calculations of turbulent boundary layer specifying merely the
change in the parameter HeeSVS**, where i** is the thickness of
the momentum loss, characteristic of the braking process. The change
in the parameter H along the length of the converging channel at
tin (Fig. 10) shows that the principal deformation of the profile
occurs in the throat region. A further small decrease in the throat
leads to the formation of a detachment profile, i.e., the parameter
H on curve 10 should increase almost vertically.
Assuming that a detachment profile occurs an the throat on the lateral
wall and taking H (x) as shown in Fig. 10, we can calculate the
phange in relative to Y. The throat area, at which :.., will be
le minimum for a given value, of po.
?
The best shape of the ? diverging channel was shown experi-
pentally to the cylindrical. For this case it is easy to determine
the total-pressure losses, after determining by previous calculation
31
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the parameters of the flow at the inlet to a cylindrical channel,
solving simultaneously the equations of conservation of flow, energy,
and momentum, without allowance for friction forces.
To verify the approximate procedure of calculation, we calcu-
lated and tested experimentaI4 channels for 14=2.5, 3.0, and 3.5.
The calculation and el!rimental results are shown in Fig. 11, from
which it follows that Fan is determined by computation with accuracy
3 -- 5%, while the accuracy of 45;u:reaches 6%.
Received 21 November 1958
BIBLIOGRAPHY
1. Bam-Zelikovich G. H. Oalculation of the Detachment of the Boun-
dary Layer. Isv. AN SSSR, OTN (News of the Acadmey of Sciences
USSR, Dept. of Tech. Sciences) no 12, 1954.
3?
Figure Captions
Fig. 1. Diagram of experimental setup.
Fig. 2. Flow pattern in converging channel; ---- -- calculation
flow of an ideal liquid, 0--C) -- experiment.
33
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2.1
20
18
-16
14
10 17
' IPS
ISM
RP
110111- dX' M
Fig. 3. Comparison of the flow in converging channel with
curve a of critical pressure ratio; the points b and c correspond to
P '-'P2/P1 and r-P4/P3.
?
i I
"4144
1%0.55
i 1 !FA
1.0.71
... .
u
2-A7.
10
Fig. 4. Change in velocity profile in the throat with changing
throat area.
34
; *
Fig. 5. Change in the velocity profile along the length of the
channel at Fmin(zl.O corred?ondsA0 the section of the diffuser
throat). .
e e
Fig. 6. VariktiOn of 41orit the sidewall in the throat with
changing F.
35
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0.016
0.014
C..CS5
0.01214
18
11 75
Fig. 8. Values of 3obtained in various channels at Fatin.
51 c 1 T I
0. r411.-
f
044 i
.76
0 1004
05 OS
Fig. 9. Influence of the Reynolds number R on the characteristic
6-= f (F ) ; 1 ? . =f (PC), 2 ? ..? (PI for R= const, Po =4.55
atmos, 3 ? Crt:f Tilt at R=var, P05 to 2.42 atmos.
37
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6
S.
4.
3
N
n4
171 g
.
-.,
1
(
1
t
-fis7
t
.
--114.9
I
yz..
H
La
Z5-
30
Fig. 10. Dependence of 1.1=f (10 for power?law profiles; dotted --
experimental values.
05
0
0?02.
6
?NS.
1.1:30
.?
i
d
PH(
I
I
I
ii
'4E?
a+
Fig. U. Comparison of the results of calculations (solid lines)
with the experimental data (dotted).
38
Siock Tdbes for Measuring Drag Coefficients of Bodies
in Free Flight
Izvejc.Akedamli Nauk SSR Ot-
aye
retaMigirrigrainos ami [News
1715717-eadeeef ences
/le-avant of Tedhnioa1 Sciences, Mechanics
'and. Machine Building], No. 2, 'Mar-Apr 1959,
IMoseow? pave 188-190. ,
Yu. A. Damvev
G. I. Miihin
A brief description is given of ii,114.10f_*..- ; tube 4 meters
long with four stations far plotting the space-time variation of
fIyinkties. The setup makes it possible to measure the idiegt
cceffic nte and mimultaneously photograph the spectra of flow of
various gases over high-speed axially-w.mmeical bodies.
1. An investigation of the flight of bodies
under condition of maximum approach to natural Cat be carried out in
ballietic (shock) insta1lations/1 -- .14./. In a closed polygon, it is
possible to create me desired atmospheric conditions and to vary
independently the similtrity criteria such as the. Mich =gibers and
the Reynolds numbers over the widest possible raege.
?
The use of various gases in a shook tube makes it possible to
establish the role of sero-physical parameters hA)/kT -- the ratio
of the characteristic temperature of the gas to the impact tempera-
ture and War-- the ratio of the characteristic dimension of the
body to the width of the relaxation region (u is the gas velecity,
T:the relaxation time), during the dynamics of the flight. Pig. 1
shows an overall view and Fig. 2 shwas the diagram of a shock Ube
for the Leasurement of reeietance coefficients. ?
High initial velocities of the, bodies were accomplished by
shooting from a rifle one .of 14.5 an .using large
batches of special powders aida corresponding fasteiling of the
butt. To soften the blow during, the shooting, the rifle was placed
on slides permitting it to recoil. The sound of the shot wee-
reducted b; placing the butt in a vacuum tank 2 through a ribber ?
seal, making it possible for the butt to move horizontal The
tank was evacuated by meansiof torevaduum pump 3 of type VN4 to
a pressure on the order of 1 =mercury, controlled by,,means of
manometer 4.
Steel, duraluakinum, magnesium and bakelite balls 9.46 am
were shot with the gun. The bails were. pressed in wads
made of delta wood. At emali-velocities? to facilitate the
removal of the bells from the wads, the latter were cut along the
diametral plane.
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3?
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The outlet froi,the vacuum tank and the entrance into the shock
tube were covered witn cellophane films 0.04 mm thick; such diaphragms
produce no deformation even in the case of magnesium halls.
The ohoc k tuna consisted of four sections of
a total length of 4 meters. The internal diameter of the tube was
300 mm, thus guaranteeing absence of the influence of the walls on
the flight af the bodies.
Three sections had two rectangular windows each, measuring
720 x 100 mm, located diametrically opposite, and two round flangus
each, 150 cam in diameter, to connect the pump, the manometer, the
vacuum meter, and the gas inlet.
Before entering the polygon, the body passed through a skai
shield, an angle sector 6, and vacuum chamber 7, serving for rapid
removal of the air from the surface of the body during shooting in
Various gases.
Before the experiment, the tube wee evacuated by forevacuum
pump VN-1, 8, to a pressure of approximately 10-2 mm mercury. The
pressure as registered with a thermocouple vacuum meter 9. The
gas was then let out of flask 10, and the pressure and tempera-
ture of the gas were measured by manometer 11 and thermometer 12,
located near the trajectory.
The rifle was triggered by 26, using a signal received
from the gun'-control panel 24.
2. The values of the drag coefficients were calculated
after measurement made with instantaneous photographs of the
positions of the bell along the trajectory, at known specified time
intervals. Such a method of plotting the space-time dependence of
the flight of the ball was adopted by us because it is easier to
obtain by electronic means calibrated time intervals, than to measure
the time when a body passes the fields of light beams with the same
accuracy.
Simultaneous photography of the sphere and of the coordinate
rule were made with tverAmaeleiriga4 light transmitted through plexiglas
rectangular windows 13, by means of cameras "Kiev", 14. To obtain
clear photographs of a body moving with a velocity greater than
100 meters per second, it is necessary to have exposures on the
order of 0.5 x 10-6 seconds. Mechanical shutters cannot produce
such short exposures, and therefore, prior to the firing, the lenses
of the cameras were opened by means of relay 25, and the exposure
time was determined ticamxiism by the length of the light flash.
40
-
an.
???
0
.1.7.21C
Illuminating apparatus 15 had a system of mirrors 16 and
condenser lenses 17, insuring illumination of the entire field of
aach station from a single source 18.
Transparent coordinate rules with millimeter divisions were
mounted inside the tube below the.' .flight trajectory. The setting
of the rules was checked by means of .a 1-1/2 meter beam compass.
The cameras were placed relative to the trajectory in such a
way, that their fields of view overlapped. In the case of deviation
of the ball from the mean trajectory,
the photograph of its position relative to the rule was ,
seen displaced. This displacement is small neer the axis of the lens.
The position of a ball that deflects away from the axis of the lens
is recorded simultaneously by two cameras, making it possible to de-
termine accurately its exact coordinates by simple computations.
Correction for the taper of the'rifle could also.be made by measuring
the deviation of the ball at the exit from the polygon.
It must be noted that in many experiments the deviation was
less than 1 cm, and consequently the correction was necessary only
in rare cases.
At the first station, which had a round field maks
of 150 =diameter, the procedure was not only to mamma the coor-
dinate, but also to photograph the shadow spectrum of flow around
the ball. .A condenser lens and a point-source spark illuminator of
the type "cylinder-electrode* 19 produced a parallel beam of light,
which was projected on the .ground surface of a
parallel plate, plate, wtichwas provided with a vertical coordinate ref-
0 arenas. The scale of the photographs was determined by first
photographing a millimeter grid. The measurements were carried out
directly on the negatives using the 1MR42 measuring
microscope, which has an accuracy aft0.3 me.
The met suitable circuit for spark production was one em-
.
, .playing a discharge of a capacitor, first ()barged
_
to 14'-- 16Av, through a pulsed hydrogen thyratcoa TGI4-325/160
Such a circuit produces a Minimum spread in time between thepassege
of the triggering pulse and the instant of appearance of the spark
m
since the proe' of initiation is lacking. The pulsed9hydrogen
thyration permits exact control, realised by applying a low positive
voltage to the grid (the time spread 414 operation, based on the rated
data, is not it worm than 0.04 x10-0 seconds, even with only 6 kv
on the anode).
To improve the breakdown conditions, pointed electrodes were
T-?
1
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used.
)
In view of the small pover'of.the spark; the glow of the plasma
was rapidly damped by the ourrounding air, and the effect of afterglow
wad-limited tam by the finite sensitivity of the film.
34.A series of pulies, arriving in sequence to the spark devices
at . exactly-known time intervals, was generated by a multi-
channel electronic synehronizer 22-0 the block diagram of which is
shown in Fig. 3. The signal b, applied to photorelay 23 when the
bullet crossed the light beam, was applied to a starting trigger c,
which controlled the impact-excitation generatoi d.
vets
After the trigger turned, the generator started *rating
with the same initialphase with constant amplitude and frequency.
In the next stage e, the sinusoidal voltage was converted into brief,
almoit rectangula;-.pulses. These pulses pass 'then through a frequency
divider f with a division coefficient 32, from which the Y Were applied
to uraivibrator g, which formed broad pulses, equal in duration to the
period "Of the master generator. -
The pulse from the univibrator separated, in a coincidence
circuit, one of the pulses that come directly from the shaping stage
e thereby eliminating the "floating" of the output signals with
time the Use of a binery-type divider with 'a large division
coefficient. -
'calibrated
The pulses from the coincidence circuit passed to a distributing
block, Which served four spark devices. After the operation of the last
of the stages of the distributing block, the starting trigger returned
to its initial. position and the oscillations of the generator were
quenched.. For the spark devices to operate with a minimum Spread in
time, pulses with en'aiplitude-of 350 volts and a current of 0.5 amp
were fed from the output stages of the synchrodiker to the grids
of the hydrogen thyratrons.
Thok-generator frequeney was controlled by 'a parts heterodyne
calibrataiRtype-NS-221T. The relative aconracy of the measurement
of the'freqdency-vai 0.1%.
-The-electroac-circuit a:limed the appearance of light flashes
at known' time interVali;- differing from each other by tenths'of a'
airosiecind;
4. The .dre:gi coefficient of a ball flying in a gas, 177
definition, equals
a
8mg
p Ed%
where mis the mass or the ball, a the deceleration, 11 the gas
deasity,ft the ball velocity, and d the ball diameter.
It is interesting to note that the time does not enter directly
into the expression for 0x, since VI and a i?-? tr2.
Knowing the pressure of the gas p and its temperature Tin the
Ube, it is possible to determine.tha density from the equation of state
p = 0.3594 po
Here is the gas density at cPq and a pressure of 760 mm mer-
cury.
The mass of the ball was determined by weighing on an analytic ba-
lance, and the diameter was meaSured with a micrometer.
The magnitudes of the deceleration and of the velocity were deter-
mined from the dependence 'of the ball coordinate on the time of flight
either by the method of averages or by the method, of least squares. The
number/41.s found after calculating the velocity of soundlvith allowance
for the temperature correction or through the use of available tabular
values of the velocity of sound. Fig. 4 shows the results of the measu-
rement of Ox in air at atmospheric preesure? using magnesium balls for
Nachrnumbers from 2.4 to 6,1 and Reynolds numbers from 5.0 x 102 to 1.0
x 10`J. The Reynolds number varied in proportion to the velocity.
The average deviation of. the measured values from the mean
curve amounts in this case approximately to
When the investigation was performed in a gas-of high mole-
cular weight, such as Freon, the mean error rarely exceeded 0.5%.
Basedcon the above, it can be concluded that the
shock tube is a convenient method for measuring the flight of bodies
with supersonic velocities.
In conclusion, I express py gratitude to A.A. Sokolov, who
has rendered great help in the wiring and operation of the apparatus.
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Cs
It
4
Pig. 4.
0
4. .0
Ps_
???????????"1111
? BIBLIOGRAPHY
1. C barters A. a. Thomas H. The Aerodynamic Performance of Small Spheres
from Subsonic to High Supersonic Velocities. 'JAS. v. 12, A 4.. 468. 1945.
2. Seeger H. .1.0n Aerophifics Research, Amer. bourn. of Physics, v. 49. NI 8, 1:)9.
1951.
3. If a y A. s. Witt B. Free.lrlight Determinations of the Draft. Coefficients "1
Sp2h?ros. MS, V. 20. :14 9, 635, 1953.
4.. Hodges A. J. The Drag Coefficient of very high Velocity Spheeres. IAS, V.
10, p. 75.1, 1957.
44
TWFORMNPION ON THE STATUS OF SOVIET RESEARCH IN HYPERSONICS
Izves?._iaAkaderiiii. Nauk SSSR,
(Meleni e Naukiki
sews of the Academy of Sciences USSR,
Department of Technical Sciences71
No 9, September 1958, Moscow,
Pages 157-159
M. S. Solomonov
/The following material is an extract translation
of an article entitled "June General Meeting of
the Department of Technical Sciences of the Academy
of Sciences USSR.5
On 16-17 June 1958, under the chairmanship of Academician
A. A. Blagonravov? a general session of the Department of Technical
Sciences was held, at which two scientific reports of considerable
significance were e:remined. The first report, by Corresponding
Member of the Academy G. I. Petrov, was devoted to the problem of
the motion of a real gas at velocities considerably exceeding that
of sound.
The rapid development of aeronautical and rocket technology has
presented aerodynamic science with mazy new and difficult problems
and resulted in increased requirements for accuracy of experimentally
obtained data. For the analysis and design of vehicles flying through
the atmosphere at high supersonic speeds, and also for the design of
power plants, the study of he motion of a gas in close proximity
4-5
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to the surface skin, where the effects of viscosity and heat conduc-
tivity become evident, is of great importance. This matter arises
because the boundary layer in supersonic flow can significantly
change the nature of the shock waves generated by the body, and this
IS especially important when flying at high velocities.
A basic question, freqnently determining the "to be or not to
be" of any vehicle, is the matter of protection from aerodynamic
nesting. The initial experimental investigations of the velocity
distribution in a supersonic boundary layer, conducted with "micro-
tubes" and besically by quantitative optical methods, showed that
the velocity distribution in both a laminar and a turbulent boundary
layer, for velocities at the boundary exceeding that of sound, is
similar to that in a subsonic boundary layer. The velocity distri-
bution in a turbulent layer is well described by exponential laws.
In supersonic flow, in re Ions characterized by a sharp longi-
tudinal variation in the flow parameters (the base of the shock wave,
floe around an obtuse angle), the fundamental propositions of boundary
layer theory fail. In these regions it is impossible to neglect the
pressure change across the layer and the possibility of transmitting
the effect of disturbances forward against the flow. Conseqpently,
the equations for the boundary layer, equations of a parabolic type,
cannot describe the phenomena taking place here. In these regions it
is possible to apply the basic equations for a non-viscous gas, but
ender conditions of mixed vortex flow.
40
7fs
Mo.
The study of the interaction of strong pressure dumps with the
boundary layer has permitted the establishment of the general mechanism
of the onset of this special kind of "crisis" wherein, at the time of
the attainment of a critic'll ratio of the pressure behind the jump
to the pressure before the jump, the shock wave changes in nature so
that an additional jump is formed; or, in the ease of emission into
the fluid with back pressure, the pressure jump is transferred into
a region of lower Mach number, so that the ratio of the pressures
does not exceed the critical. This critical ratio for a turbulent
boundary layer is a function only of the Mach number of the approach-
ing flow and has been determined experimentally over a wide range of
Reynolds nnmbers for Mach nuMbeee from 1.5 - 6.
The discovery of this effect and the obtained universal relation-
ship for the critical pressure ratio as a function of Mach number has
permitted the clarification and the predicting of a series of other
effects connected with flows in diffusors, altitude chambers, around
airfoils, around braking flaps and other cases of practical importance.
This has also permitted the development of a method of calculating
the thrust of a nozzle in the uncalculated regime.
When studying the deceleration of flow it diffusors and the exhaust
of nozzles, of fundamental importance is the study of the laws of
motion of a "closing" pressure jump, i.e., a jump which can be dis-
placed or deformed by the action of a back pressure. The relation-
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ships which have been obtained permit the determination of the
statically stable positions of a closing jump and the calculation
of the losses In ducts.
During experiments on the maximum deceleration of a supersonic
flow in converging channels by N. N. Shirokov, a nee phenomenon was
detected: the formation of a new pressure jump on the side walls
of the channe3 due to the breaking away of the boundary layer.
This determines the maximum possible increase of pressure as a
function of tne Mach and Reynolde numbers and of the shape of the
channel.
As the author showed, the numerous semi-empirical methods of
calculating the coefficients of heat transfer and friction with
supersonic flow in the turbulent boundary layer are based upon the
application of integral relationships and the establishment of a
connection between the local characteristics of the boundary layer
and the local coefficients of heat transfer and friction. These
relationships, obtained from a finite number of experiments at low
speeds, have been broadly extrapolated over a wide range of Math
numbers and ratios between the wall temperature and the stagnation
temperature. The results of calculations based upon these methods
differ widely amongst each other as the Mach number increases.
For an experimental investigation of heat transfer and friction
in turbulent supersonic flow, the working out and development of an
extremely precise methodology for the direct measurement of the
24,3
0
local coefficients of heat transfer and friction is regnired. The
experiments which have been carried out have permitted the eetiCelish-
meat of reliable criteria for the physical relationshi:,
?a comparison of the various methods of calculation. szx,istinE
methods of calculating; the best agreement with exnerimee . eee a
wide range of experimental conditions is given by a ezeeee eeeelol,e6
by V. M. lyevlev, which takes into account the moleeuier diee. let tee
of the gas.
During an investigation of the flow and heat transfer on a blunt
nose on a body flying at a high supersonic velocity, there was
established the law of the constancy of the relative distribution
of pressure as the Mach number is varied over a wide range, and there
vas also determined the coefficients of heat transfer on bith rigidly
supported and free-floating bodies. Rather high coefficients of heat
transfer on free-floating models were obtained by virtue of the
roughness of the surface at the time of its disintegration.
The methods which have been developed, together with experiments
which have already been carried out, have permitted the establish-
ment of the existence of anomalies in the boundary layer structure
during evaporation on the wall or during the injection of another
gas through a porous wall, and have further permitted an evaluation
of be effect of7 a reduction in the thermal flow which is important
when developing methods for .protecting structures from thermal effects
at very great flight velocities.
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The author of the report pointed out that the investigation of
the transition of the laminar boundary layer into a turbulent boundary
layer is still only in a rudimentary state, and we have only partial
information on the effect of the various factors on the Reynolds
number of the transition. But evidently, as the Mach number increases
beyond 5, with a relative reduction in the wall temperature the tran-
sition will be delayed. Special importance can be ascribe& to the fact
that with a high longitudinal velocity gradient, the reverse tran-
sition can take place, namely from turbulent into laminar. This vas
detected in experiments with the boundary layer when investigating
the losses in the nozzle of an engine.
- - -
;0
7-7
Flow Around Around a Conic Body During Motion of a Gas With
High Supersonic Speed
Izvestiya Akademii Nauk SSR, Ot-
deleniye Tekhnicheskikh Nauk:
Mekhanika i ashinostroyeniyei (News
of the Acaderw of Sciences USSR, DOD-.
artment of Technical Sciences: Mechanics
and Machine Buildingb No. 10 Jan-Feb 1959,
Moscow, pages
Aa L. Gonor
1. Description of the general method. We shall consider the
flow around a conical body by a supersonic Laas stream with the associa-
ted shock wave. The surface of the body is 'aiven by the equation
F(xiz, ylz) 0.
To derive the equations of motion we choose an orthogonal sys-
tem of coordinates) in which the followinG holds: a) the first coordi-
nate family is the spheres r2 x2Ary2e-e2; b) the surface of the bo-
dy coincides with one of the coordinate surfaces of the first family.
Such a system can be determined if the second family is taken to be
the surfaces F(a/z, y/z0 6 ) 01 obtained by introducing in the first
equation the parameter 6, and if the third family is taken to be the
conical surfaces AE (x/z, y/z, 90) = 00 superimposed on the orthogonal
trajectories to the surfaces of this second fnmily (Fig. 1). Consider-
ing that the conic flow is self-similar with respect to the radius r:
we obtain after several simple derivations from the general Lagrange se-
cond type equations L 1.1 the following system
r du :r
??? r2 2L.2 ? 0
ov et: 01,
2-- ill' W
+ ru. (In :tov
:12
(4i = prx02 ..7... yo2 .I. 21,2 )
r
(In AA, I rap
U.2, UK-- -----. ..?....
.al pill de
rIir p `772 r2 174 trs r () p r2 7r2 =
0 0.0
Ai a ty-1 + 1.7 ? 1 ; 2
"
Ai a if :12 d?
a PI) + 442-) 0 (A2 1(X2 M;2 + )
dp p (1..40 _,_
2.ou -I-
t ' r a ? . 12 d9 m Ai :12 L 1,9 I
The first two eouations are the projections of Euleris equa-
tions alo g the axes r and e. The last three equations express the
2ondition of conservation of energy, entropy, and mass of the parti-
Aes, respectively. The functions Al and A2 are the Lame coefficients,
2alculated on the surface of the unit sphere; 110 v0 and w denote respec-
tively the velocity projections 'on the axis r: e, and 91; nl 9,
el and 4'
%re the pressure) density, and ratio of the specific heats. We convert
the system (1,1) to a new variable 01==11'69, 0 satisfying the following
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/03/14: CIA-RDP81-01043R004200140002-4
equation
VD1LI w , 0
? ;IT Tfe "It gig; ?
The surface ib const represents the current surface, and there-
fore the variable y plays a role analogous to the usual stream function
for two-dimensional and axiall,y-symmetrics.1 flow, if all the motion is
considered on a sphere of unit radius with the center at the vertex of
the body (Fig. 2). r.11.aking into account the connection between the de-
rivatives c Voe, - 4 / we obtain for the changeover to the
changeover to the new variable the following relations
0 i a e
> 1.1" t muk:.."...4.1IQfl
scr'cs in th,
P?F0+4"-1-????
v =cat,' ? + ? . . + 'not+ ? (1 s)
P?pfolt-t-pi+???,
? . .
ina,c.rting coric,z into (1.4) and (1.3) and int:- tsx.>
tIL s=fice :?.f the ?,;,3 ccuations and
bcundery fr thc ter.r.s t4,(.1 lith aqua?
2. th tcrm;:. thr
scrios, 1,74. (1.2) 1.:(- -;btnin th, "Irst trr of th
14- --Ape sele(lnAt).-4047?4
# __11 ? we] a
f. -t- -I- ? ?0,
81L..}.0
a
-4-11n (p. Ai ws11.4)1 ? At, 1.21. 42_14LI v,
iv* 42 4 AI
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53
(2.1)
I.
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Here and below the Lame coefficients are calculated for 8a a
and depend on a single variable fo.
On the shock wave, in the case of ,P-a5V*, we obtain from (1.3)
the following boundary conditions:
? ve vo"J'
Alice 20 1
243 1-1
pe. = C v0, Po' = p0 [1+ er
11? = U00,
Wo = Wo0
(2.2)
here u8, v8, and wo are components of velocity of the incident stream,
determined from formulas (1.4) for 6* as ek. Integrating the first,
third, and. fourth equations of the system (2.1) along the line le, =
const from the point 1,1 to the point N (Fig. 2), we find
U0 = Ao(41) sin A2 cPf.
w0= AO (CO COS b A2 4 + ct(tio)j
The arbitrary functions AD, O,and
equal to the following expressions
221=a0m (2.3)
Po
from conditions (2.2), are
.60 (4.) = _+_. 1/-(u0c12 + (zvo?12, a (4,) --a arc Ig
2a2 1
60 = (7.0")2 [1 + (y ? 1) vo?':
(2.4)
'The sign in front of the square root should be the same as the
sign of w8I. The primes here and henceforth will denote that the cor-
responding parameters are calculated on the lines of intersections of
the surface of the shock wave with the surface of the strewn v----
const (the point N on Fig. 2), when q r. It is obvious that a unique
mutual relation =lots between the variables and. r and that the
sought functions are best determined in the en.riables pi and fr. The
fifth equation of the system (2.1) after deteneralnatJ-)n. of uo and Iso;
admits of the following integral:
po' Al' 0,14' po Ai 80+
woo,
WO
(2.
This relation makes it possible to represent the second equation
of the system (2.1) in the integral form:
(la At) c wo A ? A ?
P ". Pa. . o
1 r0 ?
we' ail
1
. sof
To eliminate the unknown function acts' from under the intearal,
we changeover to an integ,ration variable rasing the inner quality
r;103. le?0%. iv?' Z13 1
e(14141. -27704-7' Li ?
(2.6)
obtained with the aid of the last eauation of the system (2.1) and the
second condition of (242), Inserting the values of f6 from (2.2) in-
to expression (2.6) under the integral sisoa, we find that the pressure
et any noint of the stream is determined from the formula
pan A rjo voaf
Po = Po p 'ST Ar'dri' (2.7)
e',41
?
Izatearating (2.) along the line 19 const, starting with the
surface of the boc..1,y, we get
4
? f nz A'd' (2.8)
lai 2 ?
Fr= this, putting f 4-2 we get the equation of the surface of
a shoes. wave 6* ( ). .A.fter integration there appears in (2.8) an are
bitrary function rigi:V)--1() at 80 0, which should satisfy, by virtue
of the boundary condition on the surface of the body, the following equa-
lity
Fig. 3
which admits the following two solutions:
61.? const
1) Following Ferri /3/, we asstree that on the surface of the bo-
dy 0- :hen. V'eouals a constant number., the value of which sde...
temined, in each specific case.
2) Aosume that 4ft/eLit?i: 0- In this case yoris found from the
implicit equation
Iwo` WI
A arctg [ tw,)-
,rd
An analysis of the integral (2.8) shows that the function can be
considered constant only when Eq. (2.9) has no solution, and in the op-
posite case the first solution leads to negative values of ec.
55
514.
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:et us return to fcrmula (2.7). The pressure on the surface of the
body is deteesooe6. from this equation r Let us investieate the
sios ef Aaa, aetermines the oiee of e-e eecend term in Ye
have
dt
:tt ,
jfq I A:, / 12. 2f ?42}
ie a anft (*.erected a1on6 theron .e.t to the line of leeeo.;eetion
c.f. toe ur.f.c., 4iti: a sphere e: *..iius; t[xol yt,
eeehel to t- n: eirected along the ee.eeee es the line q , c t the
sphere. -.Ise el of the scaler pro(luce coioeLlas with the 3L uf
) ?-1 t), wh.Lee ie pooitive (Fig. 5)) if , 'eon from a conose eee
neeetise if .hss feoo a concave surfeee. Thue, the pressure on son-
vex Deets of ths eurfoce is lcwer 'shah 'oehia- toe ehock wave, n, e. the
cere one it ie The increment dole to the second tee:- in
cooraeteriece tse iofluence of the cell-bet:au:al forces, due so trs, teonL-
veres flo cf the
Lolution of the problem of flow around a flat trtaoeoleo ,eLet
As an exaleple cif toe application of the General method, we cao eooeir
tos :Ilow about a triansular fiat wins wish o vertex angle 2., .e eheese
a .eestex of eocroinates satisfying the requirements indicate in sectien
trchler:, we take for this eystem the axec, r, r% 11. shoen in
Fib. L# The eurf.sce of the wing in this oystam is given by :,he eura!ition
C, aed the ceonectien between the ola onL new coorainatee i; Ieter-
mined by the following relations
uo,
acc
2?? cr-IS si (It
The :Lose: coe2ficients are Ai
vg este re calculated from
s
oraanee Yito Fig. 1 and with
i's fro: se....oea .o? ,?
meees " s,.e, ,e.s'
fero
y =-- r II. r Ott, 0 Cciti.li
1, A2 = cos 0, :;onsider?.ng th.t
sobstieuting for st in
CI we find the streao pares
ana 2.8) in the follee:ine
1,4c =???? 1./ OA a cos 5-, wc Ct cp,
112 ,=--71/2 [ I s -
e, 1,w-,scnj
2
1 V'
80 n cc (stg Ctg cf.:*)tg [1 ,
ts,
3y direct substitution we verify that 7e. (2.9) has no solution
for the function if?4 V) and consecuently Veis a constant, which is
found unique:le; if the shock wave is not deteched (the front edge io eu-
personic). L.scsualIy f? is the coordinate of the surface of the flow,
adjacent to the surface of the wind, and since this surface intersects
the wave on the front edge (right or left), thenr=s1p (Fig. 4). Let ne
study the flow o'eout the wing in greater exactitude by using the second
terms of the series (1.5). The system (1.2) and the boundary conditions
for the terms with index 1 are analogous to those considered in section
2. As a result of all the derivations, which we now omit, we get
sirsSin fi ? (y ? i ) Alt SIII2 1-1 4111 ? T41
2 tg [t 2
t
rr M bill-
r. MD al (7.1
C?S ? 7")
ga A
? 1) 312 a] ?
2 ?
= slit/ a 072 siill -- 2 /
the- --
? 2 4- (y 1) /1/2 slit= 7
2
03 Dz' bill tg z [1 +
I ? - a.) sin 9' I ? 11 Ali sill, 2 '
II ?
I g7 [1 + -aS [1.?2 Ct g tjj ji
(3.3)
The surface of the shock wave is obtained from the expression
atiP-4. Let us consider the intersection of the surface of
the wave with the planes z 1. Going to'cartesian coordinates by
means of forleulas (3.1), we find that in the first two approximations
the intersection line is the straight line
. ? 1 1
y ? tg I -1- (7 __1).410--c-(1
1.1 4. 1 (I ? 2
7 ? i ? I) tia:?urt a sill/ EsI (y DAP sipla-M
Conseeuently, the surface of the ehcch wave consists of two pla-
nes and has a an at x t O. The flow lines on the surface of the wing,
as shown by calculation, are straight lines that converge towards the
symmetry axis at an anglees*E. We note that the solution obtained con-
tains) in the second approximation, a singularity of the source type
(.1/-0) at the Doint of the kink of the shock wave (x m 0, y y*). It
must be aseateed that the kink in the wave and the singularity are due
to the approntnate nature of the method and will not exist in the exact
solution (see 3ection 4 on this). We have considered flow about one
surface of a triangular wave. On the trailing surface at M400 there
is formed a base vacuum /4/, and one can assume D 0. As a result of
such a flow model, we obtain for the coefficients Cx and Cy, referred
to the area of the wing in Plan; the following* expressions
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s,
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C., = sina cc 11
{1 4 :17-.77-1 15);:feln2
A A
Cif 2Sinzat COS 04/. y I IL pp We I
is o thc.t the coef:2?12it-:::,;:. C:: and Cy thu:,
wi-, c3rre.;-:on:.:1.15 coefficients as given i.
/2, . T'ne situz.tien is analogous a:so theory /5/.
-.)t7 the coefficient Cx and Cy in this thesr;; oince
wi the Ac::::ct forzulas for a wed:;e. ::hows a co:-i_in of
thr the e,:z--;criment, carried ..odel of a trian:;t:..a.r
wi-hrhomboLl:_ a profile of 5 Dercen*:, at 8 300, M ? 6.9,
e z , taiTon from reference /6/. The solid curves plotted from for-
mula:: ?,'5.4) go into the dotter curves in that -Dart, where the parame-
ter I- = I sin 2: < 1 and the theory is no longer applicable. The &ac-
he 3 011'717 the results of the linear theory, obtained in reference /7/.
(3.4)
KW
CI
CM
211---,c
a
?
-
?
C
r
*
.00'
..--
.41
?
%
? ?
. : ..)",
?
?????
.......
....0 .0.0"
_
11
Fig. 5
a?
Solution Of the problem of flow about an elliptical clone.
A second example is the flow about an elliptical cone, the surface of
whlch is given by the equation x2/a2z2 -r y2/b2z2 -1 O. The cane of a
round cono was discuSsed in reference /8/. To plot the system of coor-
dilLates we introduce, as indicated in Section 1, the parameter I.P .
tTe
the:. )btain a family of surfaces
tg2 0 = x2 / z2 ny2 z2
(ft al bl) (4.1)
:n the region where tan-1 a the surfaces tO= const
fill the vollx:le outside the body unifornly and can be taken as a second
The third fly is found by down the equation for the
Orthor.onal trajectories to the family of surfaces (4.1). In fLnal for::
thie fsmily is determined by the equation
(x/X2'1(1+xa +
210427-)1'
teq (4.2)
3y direct verification it is easy to check that the new system of
Cocr:tina*:os e;+' r is orthogonal and that when n 1 it goes into the
58
3
L1.4.1k
V'ON
ordinary sperical system. The coefficients and A2 can
be calculated from formulas (4.1) and (4.2), using the fol-
lowing relation from vector analysis
1,(467: -r Yot X02 -= fixa eY2. "1". (V.
After carrying the various calculations, we find
9
=
+ (1 ? yig) WO see d ?rojnyi2jfa:
2Vc
12 Sin 24; Vet, tom (1f:ffitt981)Ljec4-.(11-11Y11AL "+(1-- r:9+tg:1--,)-3:41? Mt
req [see@ 4- (1 ? Yi) y1211-11 Pee (4.3)
As can be seen from formulas (4.3)2 a unique relationship
exists between yi = y/z and f' / and since yi is a geometri-
cal coordinate in the plane z = 1, it is more convenient to
determine in terms of this quantity all the sought func-
tions. !Ile velocity components of the undisturbed flow are
found from (1.4 )to equal to
cos + sin a yi
ilt7=t 1.3 ? 9
*C2 eig ? 01) r4'
C" a teek? Sin a ny,
v ? U
Ltgl 0 sec.2 ex?n(1 ? n) y,21.4
(4.4)
fain a'sel 2 Ok ? cos a (1 ? n) (V ok
(te Ok sec4 k (1 ? n) !tie 8 -1- tie Ok (t n) ? n (1 ?nly24).
Fig. 6
01
92
C -
la i
ct/b.Y2
a
*11.4
liras
d
ig"
11111 III
ainiliMilk.....11.1111.1.1r.?.*
liONOMMINIMIN
airm-mww=iiimul
30
14
is
9 92 94 Q1 911
Fig. 7
iiguras 6 and 7 show plots of the distribution of
the coefficient of pressure Cp over the surface of ellinti-
cal cones at M =00 and cx = 0.
On these figures
a,b a 4 2 1 1 1
:= Y1 if16; ' ' ' ? '
/ 1.16,3.1(1-
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An investigatioa of tae derivative (90*)9 of the in-
tegral (2.8) shows that the shock wave, no matter how small
the value of a/b 0, has no kink in the plane of symmetry.
Consequently, the singularity that occurs when a plane tri-
angular wing is placed in a stream does not appear in all
the cases when the thickness of the wing is not zero.
In conclusion, the author expresses deep gratitude to
0-, G. Chernyy for great help with the work.
Received 9 June 1958.
BIBLIOGRAPHY
Nochinoli. Ye.; Ube)); N. gad RozeiN. V. Peoretichesk.vs.gidrome-
khanika(lheoretical Apiromeehmdcs),vbi 1. 3947.
2. Cla.erny7, G. G. litiow of Cellos About Bodies at High Sunersonic Speedz,
DAN SSSR, vol 107, no 2, p 221, 1956.
,7
Ferry,A. Supersonic Mow Around. Circular .0ones at Angles of Attack,
Mak T. R. No 1045, 1951.
4. Coodhek, I. Aerodinsmika exemkhzvulamrkh skerostey (Aaroctimamics
of Liih Speed.), BUS . Trenel. IL, 1954.
Z. Nrankell. P. Ye., Xernerich To. A. Greaodincatika tonkikh tel (gas
14yllmic of Slender Bodies), 1948.
G. McLellan, C.- H. Zscploratory Wind-Tumael InTestigations of Winds
and Bodies at H..6.9. JAS, No 10. 1951.
L Gurevioli..M. 1, Lifting Force of Sweptlitack. Wing, in a Supersonic
Stream, 11414 (Applied Hatb, and Mechanics) vol 10 no 4, 1946.
8. Genu. A. L. Flew Around a Cone at wa Angle of Attack with hiel
Supersonic Spelled, , He 7, 1958.
60
0
Calculation of Axisymmetric Jet Nozzle
of Least VeiRht
Izvesti a Akademii Nauk SSR, Ot- L. Ye. Sternin
delenixe Tekhnicheskikh Nauk
MekilaUlicA-LiaaatIMEIN2X21.11Y.21sews
of the Academy of Sciences USSR, Dep-
artment of Technical Sciences, Mechanics
and Machine Building, No 1, Jan-Feb
1959, ioscow, pages 41-45.
The problem of the optimum contour of a jet nozzle
was solved only in the last few years. In 1950, A. A. Ni-
koliskiy /1/ proposed, in solving the variational problem
of gas dynamics, to calculate the aerodynamic forces ap.
plied to any surface, in terms of the parameters on the
characteristic surfaces that bound this surface.
In 1955, G. Guderley and E. Hantsch /2/ gave a so-
lution for the variational problem in a nozzle of least
length with an angleentry. The authors of the work have
redueed this problem to a numerical integration of a sys-
tem of ordinary differential equations of first o:der.
An analogous problem was solved, with a much more
rigorous mathematical foundation, by Yu. D. Shmiglevskiy
/3/. In this paper, unlike in reference /2/, an effective
m04q4 was given for integrating the system of differen-
0014f4quations. In reference /3/ the investigation concer-
ned a nozzle with an angle entry and a fixed length and
diameter of outlet section. The working fluid was a gas
with a constant adiabatic index.
The calculations have shown that the minimum length
nozzle is not the best as regards weight characteristics.
It should be noted that in an exact statement of the
problem it is very difficult to solve the problem of the
maximum-thrust nozzle for a specified weight, since in this
case it is impossible to use the aforementioned principle,
expounded in reference /1/.
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On the other hand, it is natural to assume that wheri
one goes from one extremal* nozzle to another, close to it,
the weight depends principally on the change of the end di-
mensions and depends little on the specific nature of the
change in the intermediate points. In the present paper,
the weight is approximated by an arbitrary function of the
end dimensions of the nozzle. In the case when the nozzle
is stamped out from a conical blank, we obtain the exact
solution of the problem.
ghen solving the variational Problem, it is not lo-
gical to consider a nozzle with an angle entry, since in
real engines it is necessary to round off the point because
of the presence of technological difficulties and dangers
of burning.
The formulas given below are correct for nozzles with
rounding off in the critical section. Nozzles with angle en-
try are a particular case of rounded off nozzles (the ra-
dius of round-off equal to zero).
' Using calculations performed with the formulas given
below, it is possible to conclude the advisability of any
particular degree of rounding off.
The symbols are as follows:- x, y -- rectangular sys-
tem of coordinates, the origin of which is in the center of
the critical section of the nozzle; p -- pressure in the
stream; 9 -- density; pc' -- counter pressure; w -- velocity;
6 -- angle of inclination of the velocity to the x axis;?
-- angle between the velocity and the characteristic, a* --
critical speed. .
tt''' dm;
clg,
Ct.
??=1 y nuiL 512119211
sin(ct-t- 0)1
(41
coo
? ?to +
sin a
fst
sin (a. ?6) '
ctg(a +0)
511$ ? ctg (a ? 0)
*By extremal nozzle, we understand here a nozzle having the
maximum thrust, constructed to specified diameters, and
lengths, (the statement of the problem of reference /3/).
62
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e ease
V
9
1, Variational Problem. Let us turn to Fig. 1. To
the left of the characteristic AM, the flow may be arbitra-
rily vortex free, but must be known beforehand. The charac-
teristic A14 does not change during the variations. In view
of the fact that the variational problem has a degenerate
character, the number of equations for an arbitrary charac-
terictic AM and for a specified weight exceeds the number of
unknowns, and there is no solution. As will be seen from
the following, the solution is found only for one of the cha-
racteristics of the family AM, AlMi, etc., which for a spe-
cial.weight we shalll call the characteristic AM.
The thrust of the post-critical portion of the nozzle
is
x
4
Fig. 1. Diagram showing arran-
gement of characteristics.
P )1 A? 2w d( i4)14)
where R is the thrust of Sn.0A.
From the condition of equality
of the flow through the charac-
ter tics AC & CB we have
A
Q = Shdy Ozdy 0 0.2)
the weight of the nozzle is
Here So is the weight
S is any continuous function
of the surface of the nozzle
end dimensions.
Furthermore
G So + S Ey (A), y (B),x(B)? x(A),0441
(1.3)
of the portion of the nozzle AB,
that determines the dependence*
and thickness of the wall on the
a ...4
(Li)
C C
Along AC and CB, the following equations are satisfied
(see, for example, reference /V):
ctg da) te _a_ 0 sin ct sin ). 90 sin a sin ?
g1(?'14"7111 Yj -1- dy ;17r "7- y $in (a + Oj = ci
7" (Ty ysiii(a-+ 0) " ?1?"
*The paremeters So and S may include a friction factor, pro-
portional to the surface of the nozzle.
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on CII, and analogously on AC.
92(),)1A,d3-lis;sY)=0
The functional is represented in the following form
A
11. P miQ ? b1 (y) p dy Sb2-(Y) dy (1.7)
whem ml and m2 are constant Lagrange factors, and bi(y)
and b2(y) are variable factors.
We note for what is to follow that
A
8 IWO 82.4 '-=1
At the point 0, the variations are connected by the
relation
(1.6)
when'
8).c =-- (21Ac? dirktilco)4C?
OV
0.1 clh
iy) dVLCdYC
?
6)-c 40 51(c
We arrive at the following equations
f1- in ity, m2fAs bigt;.?
221.ruil-f-htirso
_t_ 90 sin a sin 0
dy y sin (a + 0) ?
s'
-
21Tit rntiz 'nits + Mt ?
0
U
= 0
at
on BC
point B
(1.8)
(1-9)
(1.10)
(1A1)
(1.12)
241>. + ;71 ti1;4 nilf.$1k+ 131E1 (Y) ? at pot-nt. C
Ilb.er(i 5 1(y) is a certain definite function of y.
We note that the Eqs. (1.9), and (1.12) fully
with Eqs. (16a), (16b); and (161) of reference /2/,
Eq. (1.11) is an analogue of Eq. (164.).
It was noted In reference /3/ that a solution of the
agreE.
systEm is the value
bi ? (1.14)
64
? r,_ .
Actually, if one assumes the condition (1.14), the.
expressionsobtained from !Cos. (1.8) and (1.9) for 0S(y)
and 64 (y) satisfy the Eqs. (1.10) and (1.13) for all ml and
m2*
To simplify the system (i.8) 4- (1.13), it is neces-
sary to substitute the values of the derivatives f'1)% ,
ft2h, 1t2A,, etc.*
After performing the transformations, we arrive at
the following formulas
- cos cc-? 2ww cos (cc ? 8) = 0 along BC (L15)
mt.+ 2?cypiv2-tg sin28 -= 0 (1.10)
P Crg ? sin 0. C.05- unsinle at point B (1.17)
Before we calculate the parameters in BC, we must
find x, and y at the points BC and the Lagrange multipliers
mi and m2.
k'?
To find these ten unknowns we hay
tem of 11 independent:equations (Eqs. 1.
points B and C, four equations (Eqs/.17),
1.2 and (1.3), the condition (1.4), and
the point C of the type
ec (0.11 d (C) = a C (.0.1
which express the fact that
e the following sys-
15) and (1.16) at
the coupling (Eqs.
three equations at
?..
(1.18)
the parameters at the point C
must satisfy the equations of the characteristic AM.
It is obvious, thus, that for a given weight the so-
lution can occur on some one characteristic. This fact is
a
consequence of the fact that the system of differential
equations (1.10), (1.8), and (1.9) obtained was of the first
order.
Since x does not enter into the equations, then in
practical solutions one must deal with a system of eight
equations, which can be readily solved by the method of suc-
cessive approximations.
We note that in the arguments given above we did not
*The values of these derivatives are given in reference /2/:
formulas (19a) -- (19f). It must be borne in mind that in
formulas (19a) and (19b) of reference 2 the values sin 9 cos
9 are erroneously marked sinoccos 9.
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use quations that relate the velocity with the density and
pres ure. Therefore, the system (1.15) -- (1.18) gives a
solu ion of the problem not only for gases with a constant
adia patio index, but also for gases with any connection bet-
ween the pressure, density and velocity.
2. Concerning the Condition of Transversality on the
tree End. Eq. (1.17), found by variational methods, can be
obta .ned by another method, the idea of which, as applied to
the eotimal nozzles of specified length, belongs to Busemann
/2/.
Let us imagine that the contour of the nozzle is made
in t.e best possible manner everywhere with the exception of
the ast element (16. ie Shall 'vary this element, postulatins*
a ma iuum thrust. The thrust.of this element is
dP 2ny(p?pldy
According to Fig. 2, we have
dS S.Ii'ufIdY -4 Si c1L, dL =4x dy ctg
0P. 2ny dS P
Sm(i)-1- 4.Ct1 6
Hence
? ? 4??????1:?????? ? ??????????
(2.1)
2.3. 2. For use in the derivation of the condition of
tram Iversality on the free end.
In the variations of 6 2 the factor in front of the
fraction is constant; furthermore, when the Mayer flow is
form d, A = const. It is therefore enough to differentiate
the .ndex (2.1) the fraction with respect to .,u. and to set
the .erivative equal to 0.
Cons.derin5:c that
18(1 d p
FLP tg x
66
(2.2)
?
obtain after differentiation exactly Eq. (1.17).
3. Particular Cases. *Nozzle of specified lensth.
7: tnis case
T (So + 1-)
? .era a" is a dimensional factor. Let us assume j-=
? : obtain
O '
0,
3y virtue of this, Eq. (1.17) bedomes
211.-.71f cq; sin 2.b
this equation is given in reference /2/.
1 and
ozzle is stamped out of a conical blank. Here
C - o DI (I I ) ? Y (A)J 1- DI (B) y (A)1
Hence
?
L24 2.% (8) N (i)?yeA
y-12.4. tv (B) y (AV-
L [If p) (A))
31. itt teLo_+ ty (8)? y ON'
S;(? + 2y (Brry (8) 31,(4))
ly (81 4- y tA)1
me transversality equation (1.17) becomes
P otga? %in 0 cos 5412 0 V- 2m (Mb (B)? V (An
LIV(0)?V(AV
4. Problem with je_lshIgalyalent.. In most practi-
ll cases along with determining the optimum nozzles of a
_yen weight it is necessary to choose the degree of expan-
_oa of the nozzle. In many cases, one is guided here only
' the weight equivalent, i.e., by the number 1 , which shows
many 4ilograms of nozzle thrust are offset by one kilo-
of weight.
If is specified, it is advisable to solve the va-
^.ational problem concerning the minimum-wave nozzle in such
? way as to obtain simultaneously the degree of expansion of
t:e nozzle.
3
In this case, the expression for the functional beco-
67
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P ? ?mQ kgidY b:MY
:aturally, the system of equations that express the
solution of the problem remains the same as in Section 1,
with the exception of 61:a. (1.16), This is replaced by the
followinc! equation
27:upw2tg sin 5L
The number of unlimowns is reduced by one, since the
Lran,se multiplier M9 drops out. Simultaneously the cou-
lLa' (1.5) drops out of the system of equations. There
remain seven unknownstA,e=": and y at points 0 and 3, as well
as the unxnown ml.
Renar'K. .:hen making the variation, the form of the
initial Tportion of the nozzle was assumed artitrary.
Ho.:?ever, by virtue of the fact that the characteristic AY1
is 2Ixed, the 3eometrical characteristics of the section CA
did not enter into Eq. (1.15), (1.16), and (1.17).
(4.1)
Nevertheless, in
usin7, for oxample, IsAs.
ry to take into account
entr7.
a numerical solution of the problem,
(1.2) and (1.3), it becomes necessa-
the parameters on the line OA.
Let us for.aulate the solution for a nozzle with angle
,
Let the-line OAB be an intermediate stream line of
the solution.' The line CB will be a section of an extre-
mal. The flow through AC will equal the flow through CB,
etc., i.e., all the equations will be satisfied for the con-
tour of the nozzle, with the exception of (1.17).
Thus, by solving the problem of optimum nozzle of
weir.;ht for a contour with an anc.ae point, we thareoy
solve the variational probleJls for each intermediate stream
line, therebj determining the extremal contours for the ooint-
2, located on the extremal. Analogous results for the exter-
nal proble::: of as dynamics were obtained in reference /3/.
:atufally, these extremal contours will not have tae
68
Received 4 June 1)53.
Declassified in Part - Sanitized Copy Approved for Release
?
.aZY;
4
2-1
1.
2.
3,
BIBLIt./GRA.PHY
Nikul A. A. On Bodies of Revolution with Channels, Having
Min!,.33_31nd Wave Resistance in Super sonic Plow, T. TeAG,I (Works
the Central Aero-Dyraamic Institute) , 1950.
G.3.c.,trley, and Hantsch, Bei' Shuzwe of Axisymmetric Su.personic Jet
Nozzles (B.uss. Trans?. i MELIthan- tr4,1 ()iechanics) 4, 38, 1956.
Shmlevskiy, Yu, D. Col tain Variatta Problems in Gasdynamics of
Azi symmetric Supersonic Flow; , 1M (Applied Mechanics and Mathe-
matics) vol MCI no 2, 195?.
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Exeerimental Investi ation of Self-Oscillations of
3cuare Plates in Supersonic Flow_
c ii laukSSRCYL.'lzvestiaAaden
l-
deleni e Tekhnicheskikh Nauk
nhhanika i Mashinostro eni el gews
of the Academy of Sciences USSR, Dep-
artment of Technical Sciences, Mechanics
and Machine Buildine, No. 1, Jan-Feb
1959, Moscow, pages 154-157
de investigate the self-oscillations (flutter) of a
square flat plate in a supersonic stream at Aach number va-
lues M = 1.7, 2.3, and 3 for the case when two edges of the
plate, perpendiculartto the stream, are clamped/ while the
other two edges, parallel to the stream, are supported.
The results of the experiment are compared with the
theoretical solution /1/.
1. peperinlagtal Procedure. The specimens were made
of steel 1 Khl8N9 (0.,;*6 80 --120 kg/mm2) and of duraluminum
D16AT (0-= 40 kg/mm2) measuring 300 x 300 mm and 250 x 250
mm, of different thickness. For the steel the thickness of
the plates varied from 0.3 to 0.8 mm, for duraluminum from
0.5 to 1.0 mm.
The fixtue for clamping the specimens in the wind
tunnel is shown in Figs. 1 and 2. It comprises a plate which
is attached with two edges to the walls of the tunnel, while
the other two edges are wedge-like, for streamlining. The
plate has a square cavity in the center. In the bottom of
the cavity are drainage holes for rapid equalization of the
pressures and to reduce the damping of the air in the cavity.
The tested specimen is secured from the top of the cavity.
The method of attachment of the specimen to the plate is seen
in the included photographs.
The front edge of the plate is bent at a right angle
and is clamped by two steel strips, with the aid of which
the plate is attached to the base plate. The side edges of
the plate bear, both on the inside and the outside, against
steel triangular prisms. The prisms are attached to the
base plate by screws. The rear edge of the plate le clamped
by means of a steel cover plate. By adjusting the screws
with which the rear cover plate and the front bearing prisms
are secured it is possible to choose such a position, at
which the edges of the place can come closer quite freely
G. N. Mikishev
70
- -
-
0
41111?311,
Fig. 1
The fixture is 16Veled (horizontally) in the working
portion of the wind tunnel.
Thus, the plate is exposed to the stream at a zero
angle of attack, from the upper side. On'the lower side, in
the cavity Of the fixture, there is stationary air.
The pressure in the cavity is practically equal to
the pressure in.pie stream.
The pressure was measured at several points both in
the stream and idside the cavity, by means of mercury mano-
meters and also .by pressure transducers of the rheochord
type.
Fig. 2
To determine the instant wheA,flutter occurs, and also
to determine the frequency and the wave form of the oscilla-
tions, resistance tension gauges were used. The tension gau-
ges were fastened to the lower side of the plate. The wires
from the tension gauges passed through the body of the base
plate outside the wall of the tunnel.
Before each exposure to the air blast, frequency tests
were made on the plate by resonance method. For this purpose
the fixture was suspended on rubber shock absorbers. The os-
cillations were excited by a directional mechanical vibrator,
which was fastened to the fixture. The resonant frequency
was determined by means of a tachometer and from the oscillo-
gram of the recording produced by tne tension-gauge trans.
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dicers. The wave form of the oscillations was determined
by :deans of sand. The only plates selected for tests in the
wind tunnels Were those in which the natural frequencies de-
viated riot more than 10 from the calculated values. .
. . .
-, . ,? .
Duriggethe.Xime of the blabte the Trpceseeeof the os-
cillations.waSalsdinVestigated latri'the?aid:Of hiRh speed
. e . . ?
motion piatUX:e photography.
_
. .
The plate was Made to flutter by selecting the thick-
ness of the plate and by continuously varying the pressure
in the stream.:4 a constant Aach number.
2, -Certain Results of the Tests. ObServatians have
shown that lona: before .the plate begins to flutter intensely,
the spectrum of the natural frequencies is strongly defermed.
For example, the fundamental natural frequency of the plate
at the instant of occurrence of flutter increases by core
than 1.3 times compared with the frequency in still air.
??????II.
Fis.
i-: 4 el
Ai the same time a Change occurs also in the wave form
of the oscillations.. or example, the profile of the Pre-
auttPr',Wave form of the oscillations of the fundamental tone,
unlike the profile in still air, is not symmetrical, and the
peak of the profile is shifted towards the rear edge. Fig. 3
Shows the theoretical pre-flutter profile of the wave form of
the.efundamental-tone oscillations.
The actual profile, as shown by measurements, was suf-
ficiently close to the profile shown in the figure.
, In the stability region, there are observed weak oscil-
tiOnb.ot the plate in the stream. These oscillations occur
a.:'the-natural?frequency, have a random character, and are
Da*Yidly''daMped. :Then going during the boundary of the stabi-
lity region, the randomly occurring oscillations are replaced
tei intense flutter.
eee
' -Fig. 4,shows curves .of the process of the currents of
-f_ldtter, reCorded With the aid of strain-gauge transducers at
, ?
(the-noints 24 3, and 4 ,(see Fig. 5). First the oscillations
:that eccUr,bepause of various random dlsturbances in .the .
'stream are raPidly damped (Fig. 4a, b)e Then, 'as'the pres-
sure 1n thestream is increased, they are gradually changed
72
Vcvi wv ?..?VvvW-v.\i,
20.04 see ?
Transducer
-rnieshAW/Womv00~40.00.4WWwoom
Traasducer 3 ?
kivoimimosierweme. AVASSMIV#0,~6101010MotWANoft?
Transducer 4
4??'"vo"'""'"'"'"*"??~"1040"*"""ft ww."1?444111"1"."1""",""'",
a
v.v V v? vVvvtiv Vti ?
Hg. 4
iato intense undamped osaillations (FIE,. 4e, d).
In the case of nattiral oscillations of the plates, the
wave-forms of the odcillations are standing waves, and in the
flutter mode the oscillatiOnsof the plate recall traveling
-eaves. This is seen from a review of,the filml.obtained with
the aid of high epeed motion picture-photography. Pis. 5
shows certain frames of ,this film for a steel Plate 0.3 mm
The photographs'shoW apProximately 4/5 of the length
of tne plate on the rear edge. The front portion of the pla-
te is covered by the wall of the tunnel (upper dark corner).
Tae direction of the stream is from left to right. A square
grid with a pitch of 1/5 of the length was drain on the plate,
and only the transverse lines of the grid are shown in the
photorz.raph.
The first photograph corresponds to supersonic flow
over the plate before the occurrence of self-oscillations.
The subsequent seven photographs fix the positions of the pla-
tes also during the time of one cycle of strongly developed
flAter. The photographs display clearly the motion of the
liu.nn in the plane of the plate. Consequently, the flutter of
the plate represents traveling waves.
During a certain time the plate oscillates with a
conetant amplitude. Then a fatigue crack is formed at the
0
73
0
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ONO
Fig. 6
edse and the plate begins to disintegrate.
The disintegration of the plate proceeds opposite to
the stream. In photographs 9 -- 14, Fig. 5, is shown;the
veIoPtgant of the fatigue crack andethe,disintegratibn of
the plate duringethe blasting process. Fig. 6 shows also
a ph.oteoraen of the-disintegration of a still plate 0.5 mm
tL.ken after the 'blast.
,
The g-retest amplitudes and thefa:St-en-disintegra-
tion occur in those plates in which the edge can come closer
durin,,f; the time of oscillations. For example, steel plates
were destroyed in this case within three or four seconds
after the occurrence of intense oscillations. The maximum
amplitude of oscillations in this case reached approximate-
ly 5 -- 9 am. The limitations imposed on the coming toge-
ther of the edr;es decreased the amplitude of the oscilla-
tions and increased considerably the time necessary to dis-
integrate the plate.
The disintegration always begins in the most highly
stressed rear edge of -the plate.
?
Various tested methods of fastening the edges of the
plate (particularly fastening of the front- and rear edges of
1,
une plate directly by screws to the fixture) did not change
the character of the 'disintegration.
The theoretical limit of the stability region is de-
termilled by the expression
0
75
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tem ?
(MN'-
ftzp clj
where a is the length of the plate, D the cylindrical stiff-
ness, p the pressure in the undisturbed stream, x the poly-
tronic index, c, co the speed of the stream and the speed pf
sound in the undisturbed stream, The value of the parameter
A for the principal region of stability, calculated for a
Quadratic plate and reported to the author by A. A. Ilovehan,
is 914. Piriures 7 and 8 show a comparison with exneriment
o.f the calculated limits of the principal region of stabili-
ty (the dotted curves correspond to the value /31 , while the
solid curve to :5A ). Firs. 7 shows the comparison for a
number of 1.7.
/
pi
III
r
,,,
Ell
Flo,. 7
0
2_3V
?
...
r
1.9
2.47
Fir:. 8
'AO
The abscissas represent the ratio of the thickness of
the plate to the length, while the ordinates represent the
ratio to Youngia modulus of the material of the plate. The
experi.n.ea:ftal points correspond to the moment of occurrence
of self-oscillations. Zach experimental Point is obtained
as a ,-.1ean, of several tests. The first two points correapond
to steel plates, the third point to duraluminum Plates.
Fir.... 8 shows a comparison with experiment of the cal-
culated limits of the stability region as a function of the
.:ach number. The curves are plotted for duraluminum plates
and pressures corresponding to sea level.
The experimental points were also reduced by recalcu-
latine: to those conditions. Each experimental point corres-
ponds to a plate of such thickness, at which the flutter
still occurs. In thicker plates, no flutter was observed.
As can be seen from the foregoing comparison, the
76
?
?
?
?
. . t
? ? ? ?? . 'I ? . o_ ?--
calculated curves are in Satisfactory.:agreeMent with expe-
riment.
: ? t
Received 9 :xitiale.1958..
o 4 , ? 4; ? 3
BIBLIOG
Movchan, A. A. On the Stability of a PahelMOving in a Gas,
PMM (Appl.ied Mathematice and Mechanics),yol. XXI,
1957."'
?
,
?? t? ?
? ?
,
???
?4
?
?
?
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?-?
77
1
-
?,
-
Self-Osoillati
Sowy
rif;
Izvestiya Akademii Nauk"
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stems in the Presence of
erna nf uences
Dr.
0
Sk. Ot- A. A. PerVozvanskiy
deleniye Tekhniaheskikh,
Mekhanika i Mashinostrdkenlivel, geWs
Of the Academy of Scieribes USSR; Dep-
artment of Technical Sciences, Mechanics
and Machine 3uilding7, No 1, Zan-Feb
1959, Moscow, paces 158-.161
r-,
The assumption of 'elowness-Of the variation of the ex-
ternal disturbances that act on the system which enters in-
to a self-oscillating mode has made it possible to develop
a sufficiently effective procedure of dynamic calculation /1/.
However, it was assumed here that the influences them-
selves are specified functions of time. Therefore, it is of
certain interest to develop a procedure for the case when the
external influence is a stationary random process, specified
in terms of its probability characteristics
Let us consider for simplicity the dynamics of a sys-
tem, containing one nonlinear inertialess element
QOP)z+P(Asr-N(Ss. u=10) (1)
?
where Q(p), P(p), and N(p) are linear differential operators,
f(x) is a single-valued odd function and z is a stationary
normal random process.
We assume that in system (1) there can be realized at
z = 0 a self-oscillation mode, and we assume that z(t) repre-
sents a process with zero mathematical expectation, while the
variations of z(t) 'within the limit of the self-oscillation
period are insignificant, i.e., with a probability close to
unity
I4si-IT.t
(T.21\
410 j
where T is the period of the self-oscillations.
(2)
'Jet shall seek a solution of (1) in the form of a sum
of periodic component xi, yi and a slowly-varying (in the
*The principal idea of. the procedure detailed below is
pointed out in reference /20).
78,
sense: indioated above) comPdn'ent
, ar.x-1-41 xi* ? itsig + ifs *. . (1)
, ?
doriiion6nts---..be ins generally -spbaki4 -
tione) of time., 1fyeV-?assume furthertore that the system (1)
satisfies certain obntit'ion&otappl?oability of the method_
of harmonic lineari.titicin,' then
: . ogy _ . (4)
and.'Wh'eiria.s'*(x0d."4-4nto FOrier-epriiii-4e,'/cali:ifitglin.
erids
Jra* 01; -x0 ?
where
;?
.ft2s it.(Ash!g+srildig
if.- '-'1.." ? i2 ..; . ... -:-....i'-: ? ? S. ?:'.., ,. ?
I ? ' , I.-. -?
??
? .;
- , ?
crde now separate frott4s. (1) the equations for the
periodic components
. ? ? .. ? ? .
?PO 211 )! 44* III CI) ? P..,!!, 'ft $4,as!,
and f oi4Thife7'sioifili Varying WoMiOnenti
Otds1.44)00411LmA, . 11.3aligesogai
Assbig-thie.the amplitude A is also a
ing 1.11notidAl.n:the sense gfj210,we obtain from the syAtem'.,
(5)-the f41-ciWing 'conditions '-, 'I
- -
' ? ? ' ,
The se4nd as usual, detirmikei-tlie fre. -
quenO'of the'selfdioillation, which in this approximation
is found to be constants and the "phase advance" effect is
not detecteA; see for :examAe reference /V,
?The .first, condition of (7) ;son be. considered._ as ;an
equittion-fbi. the-dependihoe.ot't4e-iiplitudeAomt4e,Slolfly-
- ? , ;
varying bOmponer - -
dt(A,?;00- ut
Let us assume that this dependence-ean-be-solved-.in-----
explitilt4Orm(0
%.:7!fir 1: ? . ?
79
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A?1=3 44.(311) 7._ 7.1! E. (tilt
Then the triknaker fUnetion"q6(A, x2) is expressed
only hyteritii Of x2, an4 the system. (6) ,reduces to the form
(p) P (My: 2; v* q.' (4i - ? (10)
L , . , . ' '?" ,) z-
21=1 0/ A (2.2)., 011
. An approximate solution dt.-the system (10) can be ob-
tained either ,by direct .linearization, or else by using the
method Of statistic linearization. 'We note that in the case
of direct linearization (10) there is no need for resolving
the implicit dependence (8). 'In'-fact,'we have ?
bq a? f?\'1do' 1 = re-2 ?a--;;; ;
go' "7/7, ,? .?
However, for an Arbitrary piecewiae-differentiable
non-linear characteristiC'f(x) it'As,possible to show that
e 0 when x2 - 0.
Hence
? ,
and in the expression ,for the derivative, naturally, it is
necedgary to put A ="Ad;"Where A6 li the amplitude of the
self-oscillations, calculated in. the presence of external,.
disturbances.
aq. I
96. (x2) xs zit jta.oxl
?
(H)
,The,stlaticgl,linearization ,(in the. simplest, most
conVenient'lbrid-bb.te:d,ohythe assumption Of normel,
dis-
tributi1oi ia-iydi.Y'.I.Y:!'s-cbiapbnen4 of theinPut,..Siznill.
? ;:
xis 1
exP
where is., the meanTaquaied_deviation of. *2. Then ,.
?
? ' ? . , OP: .
..?-- S ?
? ? ) = r (4'4 (..r) clx (13)
g ex F :"7-4114
t' iseasy to show tWat'upon a aialLchange in .d97/42
w -
in'the?i*oliAtile'renge'*2-b6th-methode give identipci.ii*Ults!
where* '
'
?
*It f011oWi-fi,oi eh -examination of the system (i0).that,the;
mathematical expectation x2 vanishes.
so
0
Fig. 1. Vibration accelerometer. 1) sensitive coil,
2) magnet, 3) electronic commutator , 4) contacts, 5)
Power supply.
For a more detailed acquaintance with the proposed
Procedure, let us consider by way of an example a calcula-
tion of an accelerometric system (Fig. 1). It is proposed
that the system measures the acceleration of an aircraft,
occurring during turbulaace of the atmosphere. The dynamic
properties of such a system can be described by the equation
(0ps + bp? + cp + 11) x + kif (r) les (T kp + 1) z
ja (14)
T ;27' k c 1'1 + Tk (1 + les), b Tt2 rk. 11 1 +
Here x is the angle of deviation of the sensitive
coil, z the acting acceleration, Tl the damping constant of
the sensitige coil, Tk the electric constant of the sensiti-
fer functions. -
ve coil, T24 its inertia constant, and k, k2, k3 the trans-
The nonlinear characteristic f(x) of an electronic
tions (z = 0) we have A
commutator is shown 'in Fig. 2 in the absence of external ac-
teral wind component
2he measured acceleration is proportional to the la-
z(t),_ A v (1)
In reference /21/1 the velocity v(t)- was assumed to be
a random stationary function, the correlation function of
which was determined experimentally. Forir A
Fig. 2 '71= ;47,1 P
Conditions (8 .)for this system will have the form
aor: c =0, -bob: + d + kiqi(.1, x2) -= 0
0(A. x,).-
it follows from (15 )"that
x2 A
Linearization in accordance with (11)yields
2 1 bc -ad
IT 71; x2"-= '2aki x2
It is now possible to determine the mean-square
viation 05( in the usual manner
k-2 Rit
9.0
go Itatir ksloa: +11 32(o4dto
d5
x2as S bos +b?c al+ ? a0-1-
2a
s m
Fig. 3. Correlation function
of turbulent 'disturbances (1)
and its approximation (2).
where Sz P) is the spectral density of z(t).
Let us describe a procedure for calculating by the
method of statistical linearization. It follows froia (15)
that
.4 001 Aoy Ao2 Axis) (181
and the limits of variation of the quantities are as follows
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82
141 1
21c.
v1A-1
P0,6 + a )
Here p is the gas pressure on the surface of the plate, pea
is the pressure of the undisturbed gas, v is the normal component
of the velocity of the surface of the plate, aoa is the velocity of
sound for the undisturbed gas, and xis the paytropic exponent.
105
(I)
The linear approximation of formula (1) was used in referencesA 5./1
and also in many other papers.
1. Let U3 consider an elastic plate which is rectangular in
plan and has sides a and b. We asanme that the plate is swept on both
sides aupersonic stream of gas with unperturbed velocities
directed along the Ox axis and equal to respectively U. and U- (Fig.
1). We shall assume that the plate is fastened in an absolutely rigid
diaphragm, the plane of which coincides with the plate of the plane.
The deformation sof the plateare described by the following
equations
Dv272u, as, almi Pert Pia 2 ON1) ahe
Wri arl dbe Oy
? (32.aU 8219
ox= TYE
where w (xly,t) is the normal flexure and
is connected with the box stresses in the
ing relations
(2)
f(xly,t) is a function that
mean surface by the follow-
where D is the cylindrical stiffness,. h the thickness of the plate,
E the modulus of elasticity,.. and I) the Poisson coefficient.
The sides of the plate will be considered to be' freely supported
aaw , Pt. A asw Oluo A
= 17. =0, = (4)
and, in addlt.Lon, we assume that they are elastically fixed relative
to axial disple'nements. This makes it possible to consider a con-
tinuous tranzition from a plate with the freely moving edges to
a plate whose ef-..gell ave stltionary. Let cx and Cy be the stiffmai)
coeffic.:entr, a1as:L.1c couplings. We shall specify that the
force boundary conditLons on the edges be satisfied "in the mean",
? = ????????Cy-imi rixy "--74 0
am, .
where N,,and Nxy are the mean stresses on the edges, .6x.eily.
are the mean displacements of the edges.
?
For a plaie subject to oscillations, the expression for the
normal load component has the form
sPer
(5)
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/03/14: CIA-RDP81-01043R004200140002-4
ri
Here ?o is the density of the material of the plate, Ethe
damping coefficient, lip the excess presSure of the gas. This pressure
mill be calculated using formula (1). If the plate is swept on one
side, then p = i -p4, where p+.'is determined for the velocity
aw aw Ow
(it was shown, in
particular, by Hedgepe45/ that it is possible to-
neilect the effect of non -stationarity.at large supersonic veloCities).
? ?
or/ 12 (7)
Then id , X 1 zit Cid ?,2 , X + Li3 (aw
where Mr-U+ /awls the Ivhch number for the unperturbed flow.
r
For the case when the plate is swept on both sides at equal
velocities U+= U-, the formula for-,the excess pressure becomes
simpler, since the series will contain only odd powers of
aw it.-1.tAi3faiti?..1.. A
AP 24Poti"--70- 'Tr k.7-1
(8)
,
Thus, the problem reduces to an investigation of a system of
nonlinear ?equations (2) in the case when the function q is determined
by expressions (6), (7), and (8), and subject to boundary conditions
(4) and (5). A 1.}.4rticider case of this problem is the linear trigs
boundary problem, descaibad by the equation
alw aw aw
?
Efi'V2,v 4- poil--87. 2p0hE -R- xpo)/ --,-
with bounesry condit! ons (4) or other linear .uomogeneous conditions,
to ohe).? asos of supported mounting. In this zaaur
sctairsrars-ctrcx:-.732...t..-_a; ,:orr..1a-Gion, the problem wc,s coniidered in many
papers/2 -- vitAch it was shown that at certain sufficiently
large vcd.,,es of ; +ALB trivial solution w O becomes unstable
with 37_111 disturbances. Physically this corresponds
to the oecl:rrzneo of 'nel flutter."
It will be shown below that under MOM certain conditions
the riCialinear -system has solutions other than trivial and also at
Mi. This means that oscillations of the "panel flutter" type
can occur also at M < MA, if the panels are subjected to a
suitable initial disturbance. We shell make an estimate of the
order of these disturbances below. ?
2. We shall seek a solution in the form of a series that
satisfies the boundary conditions (4)
m
w tx. 0= 2 2 lot Msinif--.Zsin *22'
.13'1 )011
107
(9)
wjere qik are the sought functions of time. Inserting (9)
into th6 second equation of (2), we obtain the function E
that satisfies the boundary conditions (5). de then insert
the expression for I (which depends on the unknown func-
tions qik as parameters) into the first equation. Applying
the Galerkin method to this equation, we arrive at systems
of ordinary differential equations
411271. dq.k
+ +f
n
?1? .I. A --/...-? 011,tigik it (qii, ...,q1? 1.1)=.0 . (10)
MA 41 G --= 1, , . ... ., on )
Here evyx are the frequencies of the small natural os-
cillations o., the plate, and fik are certain nonlinear func-
tions. The system (10) can be investigated further either
usine, known approximate methods, or else by bolving it with
the aid of electronic computers. 4e shall employ both me-
thods below.
An analysis of the correspondinglinear problem shows
/V that the flutter motion near M = M* can be described in
first approximation by an expression of the type
mr :Ty
y, I) ?-r. q1(1)sin ? sin ? q. (I) sin ? sin b
a (11)
the
for
?
Introdurqng p-xtial linear frequencies "'I and 44., and
dimensionin;_ variables Gy..; 72111==.1..1, we obtain
the case of flol% one side only
?. Z1 + + + (1,2:22) + 11.2z2 (1inzl: 1112:22)]
(c117-12 (-12:22)0 (12)
to the
-t .7 7'1;2 Z1 :L(121:1:2 (111:42 bv,t:29
Lzt. (c21:12 4- (.22221,- 0
Tae peE d,t...1,e here differentiation with respect
dir:erislonless time.
fr
4 ? ce = - -
? ' qv:1 7 EC
, Y?P?,?.? (14
7:1
A
17.a
all (x 4. ),
'2 ? :72 tx + 1.
a
--b-,
I !-
a '
:3 1 ?
11;;,volta44 (1 Ort)t
51 , 24
a1-X 1), a:1 1)
=- ? rt! (x 1), ir2z kto :72 (x 4 )
r
C21 4(1 ...?1? 0.4)
2t3 ? :2,0220,A 4
1 ? v-74,61,
2/
4. 4s:c114
Sial
(I -I- 4a2)2.
108
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???
1.
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42 ?
2Q6r 8122349v -f? refic..)
I
0 r- vt3x$y
21460,44"611A.Ayl"14130
1 1 , =.. + Eh
At; ry
rx
If the plate is swept on both sides at equal velocities, the
system (12) assumes a simpler form
Z1' [?T z2 grat(b112 -I- bi2z22)]
+ Ls, (CZ -f?c12:117.1) 0
ZIP + y% [-L; zt 4- ;121 (b12z12 b22:22)] 4-
Lz, (c21:12 cs,z22) 0
lass where K has a value twice as large as before.
We note that for the critical value /44* in the foregoing approxi-
mation we obtain the simple "formula
- R141 2
3. We ..Eccls an approximate solution of he system (13) tat
among the c....ans ct periodic motions with finite amplitudes
..i= Acoset+ BsinOti-? ? ?, 2.2=CcosOz-t-???
(13)
(14)
(15)
Fics.03 .t, CI and 9 az.,e certain unkn.ovn constants; the dots
stand for the te:-.ms. that con'valn harmonics. We consider the steady-
state se lf-osc 4.1.1a.?,..i.on mode , and therefore t he initial phase is of
no iper tano Inuerting (1>) 4..nto (13) Lad negleating the terms
that c ont Lt. larmonic s, we obtain the system of equations
(1 ? 62),1 1"811 ? !die p.3KC b,, (3.42 132) 4 b12C21
+ LA 11 (-it (.42 4 B2) -I- citC21
(1 ?99 B 15,-4 A + 4p.31k" ABCbn + LB ri en (A2 112) + (.121. 0
?02)C +-KA 11.31C A [2i- bn (A2 + B2) + b22C2.1 (1(i)
4- LC I+ r.21 (3.11 4- //2) - 114- r2121 0
43
(?? KB + 1.1.3Kll //21 (.12 -4- 132) 4. bl 01 4. LAB(' c2, 0
4 4
109
An approximation solution of this system can be ob..
tamed y.assuming that OA damping is suffioientlY.small,"
Then B2 ?icir.A?, .0 8113.e..