Development, present state and applications of localized interaction theory. Review.

Full text

Fluid
Dynanics,
V1L 31,
No.
3, 1996
DE\MLOPMEM CURRENT ST{IE OF TIIE ART, AND APPLICATIONS
OF LOCAL INTERACTION
TIIEORY. REVIEW
A. I. Bunimovich
and.d V. Ilubinskii uDc 533.6.011
Loceil interaction theory compriscs
studics based on the t€prcscntation of the fone and themal chancteristics of the action of a mcdium
on a body at a certain point on is surface as functions of thc local rclocity, thc anglc bctxEen the wlocity dircction and the normal
to the surface, and the "gobal" patameters,
which are constant at all points on the surface.
ln tbis rcvicq we will traec the history of the theory and analyze
the nain rcsulB conccraing the de\Elopment of particular local
interaction modcls, thc calculation of thc intcgal chaftctcristiG and thc detcrmimtion of optinal body shapcs, the general propcrtics
of thc modcl and practical mcthods bascd on thco, and nontnditional frclds of local modcl aprplication.
Along witb the gcncral rtsults, we will focus on thc most imporail fields of local modcl application, such as aerodynamics,
gasdynamics,
and rarcficd gas
dynamics
L MAITIEMAIICAL MODEL OF LOCAL.INIERAICTION BETWEEN A MEDIUM
AI{D A BOIX ST]RFAICE
Consider a body immersed in a flow of a medium with the free str,ean wlocity v- The problen is to calculate the
characteristics
of the action of the medium on the body surface
Mathematical models and methods
of calculating the action of a mediun on a body moving in it depend essentially
on the kind of medium (hqui4 gas,
soil, etc.), its characteristics
(density,
rrclocity,
etc"), the surface
properties and body
shape, the conditions of interaction bet$€en the medium and the body surface,
the orientation of.the body, atd other
factors. lfis 6sdsling of particular "medium-bod/ pairs is considered
in the conesponding branches of physics,
mechanics,
and engineering. In the 'classic' Eriant of local interaction theory for translational motion of the body,
problems ale considered for the folowing models of interaction betcrcen the medium and the surface:
"r=*^y_#=er(4 Ets'
* o,(a
ot'
,o=#.H#=oo(a o, €
=v-ono,
(1.1)
ei,
O=- 2
Here, A.F, LQ te the force and heat flux acting on a small area A.S
tangential
to the body surface at a giren poin!
Cr, Co are the local force and heat flux coefEcients,
4- is a pressure
head, p- is density,
n" and ro a!e, respectircly,
the rcctors of the inward normal and the tangent to the surface at a giren point; to lies in the plane of the rectors vj
and no,
a is the "globd'parrmeter \Ector, O, O., Qnare functions characterizing
the'medium-My'interaction model
the subscript - refers to the medium, and the superscript
non
denotes
the unit l€ctors.
The global parameters'characterize
the medium, the body, and their interaction process,
and their ralues are the
stme at all points on the surface. In the general case,
their ralues alE also x5srrmsd not to change with rariation in the
shape
of the body and its orieutation in the medium. The individualization of the limis of global parameter inrariability
depends
on the region of applicability of a particular model or is postulated
in derrloping the general methods of the
theory.
A local torque coeffrcient is introduced as:
-=io-r)xc,
Here, l. L a characteristic length, r is the radius rector of a point on the body snrface, ro is the rad.ius rector of
Moscow.
Translated
from Izvestiya
RossiiskoiAkademii
Naul Mekhanika
Thidkosti iGaza,No.3, pp.3-18, May-
June, 19!)6.
Origind article submitted April5, 194.
W154628
| 96/3 103-0339$ I 5.00 @L996
Plenum
Publishing
Corporation 339

the point with respect
to which the torque is calculated.
Since the heat flux representation
is analogous
to that for the scalar
characteristics
of the force action (projections
of C. on a certain choice of a:as), many results
obtained for the forces
can be extended
to heat flues. aciiraingfy, in
the obvious g3ss5
rhis possibility is not specifically
mentioned belorv.
We will call the body surface regions
where E > 0 and € < 0 "illrrminated'
and "dalkn
respectirely.
We ass,me
that the mediuqr acts only on the "illrrmilafsd" part of the body surface.
This does not e;rclude
a locai approach
to
phenomena
whose
models are meaningful in the region € s 0 since
the action qf the pedi'm on the "dark,?gion can
be considered
separately,
using the same
methods.
The total force and thermal characteristics
(integral characteristics)
are determined by integrarin-g
the corresponrling
local coefficients
orcr the "illu.ninated" body iurfa'ce.
Requiring the body surface
to be nonconcarc
is not necessary
but, along with others requirements,
can be used
to
justify solution methods in particular problems.
_ A certain hypothetical
process
of medium action on a body surface
c:r, be identified with local interaction models.
Belw, Il/E
will formulate the most inportant properties of this process
without entering into a detailed discussion
of
ffisir rcilization mechenis65- $96srimeq theie properties are referred to as the basic
iostulates of local interaction
theory.
Prupetty
7. AII the points on the body surface
interact with the medium independentty
of each
other. The interaction
at a point does not affect the outer medium, i.e. does
not affect the medium characteristics
important for the process
of medium action at other points on the surface.
Pruperty
2. tn the interaction betcrcen
a body and a medium with identical 'global" parrmerers at e\€ry point on
the surface, the ractor of the force ererted by the medium lies in the plane'of tf" tcctors vj and no, and the
quantitatire characteristic
of the force and (or) thermal action on the body is determined only by the angle €=arccos{.
Here, in particular, the body surface is assumed
to be isotropic.
The fact that a certain process
of interaction betcrcen
a medium and a body surface c"', be described
by a local
godel means only that it can be represented
hypothetically as a physical ptol* characterized by the pioperties
formulated aborrc-
Hcr*€rcr, this does not mean ilat precisely this interpretation best fits the physical nature'of the
process.
Anottrer approach
is also possible
when a particular interaction nodel is not knmm or inexact and there is reason
to beliere that it could be advisable
to seek or inprorrc a description of the interaction between the medium and the
body surface on the basis of local models. Here, the reaso'i& proceeds from a consideration of the interaction
properties to a specific model. A typical example
of such
ao apptoacn
is ffis nedsling of the intermediate ,egion of a
rarefied gas florv.
- - As the folloning discussion
shovs,
the possibilities
of the "local"
approach go beyond
the scope
of the classic variant
of the theory.
2. ORIGIN OF THE THEORY AND GENERAL DEYELOPMENT TRENDS
Particular local models hart been knoum
for a fairly long tine and are widely used
both in theoretical studies
and
in applications.
Belw we will mention some of the locai moleb most widely *"i io gasdyramics
and aeromechanics.
In the case of a slighrly_pertrubed
plane supersonic gas flonr past a tirin proflJ at a small angle of attacls,
the
models corresponding
to the first and second
approximatioos
Uetoogio the categ;y under considerati-on,
with the free-
stream Mach number M- the ratio of specific heats y, and O,=g being anong the global parameters. Higher
approximations
are also knom; in the third approxination an additional glob-al
p"r.i"t", go, "qoul to the ralue of the
angb_e on the upwinq part of the contour, uppeuts
in the qrpression
foio".
- In supersonic
an{ hyp9.t:o}" flw appliS-$oS,Ior catculating
the presiure on the illuminated region of the body
surface the classic and modified Newton models t!)6] (and their'local -"logio", for example, [1lfl-and the method
of tangential cones (vedges) [%], with rarious -eooas ior calculating
the preJsure
on the surface'or
tte i""g"itiul boay,
are widely used-
Various
modifications of the local model are widely used for describing
the force
and thermal action of a supersonic
free-molecule
florv on coil€x bodies
W?5,70,71,731.
In the case
of a parallel qgncil of light, tle action olthe luminous flu can also
be described
by a local model [104].
By analogy,
many results obtained for a rarefied gas can be e:rtended
to the case
of luminous flux actioi fSi Si S+f.
Models of'the class under consideratiol h* usually appeared as a result of an analysis of sone piysical
phenonenon (often, it ryiqg cases) and the studies bised in them ha.,re
been resrricted by the timits- of the
corresponding
field of 19chania. Up to the end of the sixties,
the hypothesis
of locelizal;sn
was
not considered
to be
a basis
for unification of the studies
made for physically
different "ooditioos of interaction between
a medigm and a body
surface
and for the derelopment of fairly unircrsal -"ihods of calculation
aad analysh of the quantitatirr characteristics
w

of this interaction. This does not mean that localization as 6 specific
property of the models had not been noticed at
all. In this connection it is pertinent to mention the analog5l
behpen free-molecular
flon 21d x lrrmlasus flux In [%],
in the course of justifying passage
from the Newton model to the tangential cone model the property of localization
of the interaction betneen the florv and the body surface
was noticed, and the models were interpreted as possible
methods of quentitatiraly describing the interaction. The paper [25], published in 1!]69
and deroted to aerodlmamical
calculations
in the intermediate regime of rarefied gas
flow, plapd an important role in distinguishing
local interaction
models as a specific category of models in continuum mechanics.
At that period, the practical requirements
of actircly
developing aerospace
sngineering called for the derclopment of effectirc methods for calculating the aerodymamic
characteristics
of bodies orpr the entire range
of flight heights. Modeling the florv past
bodies on the intermediate range
of heights lying betucen the region of Navier-Stokes ralidity and the rcgron
of fr,ee-molecule
flw was and until recently
has remeined the most complicated problem.
Taking into account
the local character
of the models
for 'dense"
and free-molecule gas
flow past
bodies, Barantsev
et at. [5] proposed "interpolatingi localization' to the intermediate region also. An approach was also
proposed
h [54], published in accessible form in LSTI blt catried out" as its authors poilt out, in 1i]68.
Despite the fact that the uqe of local models in mechanics has a rich history the origin of the local interaction
theory is associated with the appearance,
at the end of the sisieq of a conscious viewing of the modeling of phJxsical
processes
thlough &e "prism of localization', i.e. with 1trs
incorporation of the corresponding
approach
into the a priori
tools from which to make a suitable choice in &eoretical or practical studies.
Since that t'ne, such terms as locatization,
local method and theory (lav, hypothesu) of lscalization (local iilst .ti.r; harc entered the scientific rocabulary and
are nw commonly used in connection with the approach
considered.
These terms hae become usual in the titles of
publications (see references),
they hare been taken as the titles of scientifrc conference
sectiong and the state of the
art in the corresponding field has become a subject of special andysis
in reportq moaographs,
and surriela Ig, m, z
?3,33,34, 80, 100, 1031.
A mqior role has
been
pla5cd
by theratic collections
of papers
[21] deroted to results
of studies
of rarious ilspects of the theory.
As the theory has erolrcd, thei follwing tnn mupled tendencies
hac become evidenc on the one hand, an
erension of the results obtained earlier in particular cases
to the entire class
of models and, on the other han4 the
detection and study of general gorerning laws ralid over a wide range of physically
different conditioas of interaction
betneen the medir:m and the body surface
The 'recent histor/ of the local interaction theory embraces a period of only thirty pars, during c/hich a complelq
somerirnes
conflicting process
of integration of separate,
relatircly independent
studies iato a unified theory has taken
place.
The traditions of the correspeading
scientific schools,
the profile of the orgenizations,
and the scientific interests
of the individual researchers
inrolrcd harrc
influeaced the choice of specific fields and methods of study. The relatirely
independent choice of subject and the internal logic of the studies
in different fields harc resutted in the unequal
derclopment of the separate branches of the theory, which makes
it difficult to girrc a chronological anabrcis
of its
derelopment as a whole and justifies breaking it docm into separate
branches.
Below, rre gl\E a short analysis
of the derrclopnent of ald the state of the art in the main branches
of the theory.
This review does
not pretend to complete ooverage
of a problem with a list of references
running to many hundreds
of
publications.
We hale had to erclude fron consideration
or only mention many
independently
interesting studies directly
or indirectly connEcted with the problems under consideration but based on traditional models and e4perimental
methods. We hara also arnided a detailed analysis
of most of the papers,
preferring to group them under I co-moo
approach
or branch of study.
3. METHODS OF CONSTRUCTING PARTICT'I,AR
MODEI,S
Although a number of approaches
to particular models possess
common features,
these approaches hare been
dercloped predominantly for studying the intermediate region of rarefied gas florv. Problems of the dewlopment of
particular models hare been considered from two viewpoints: (i) determination of fairly general relations for the
functions O" and Q, fufividualizing the model as a basis for dertloping effectirc methods
of aerodynamic
characteristic
calculation and identifying particular model par:ameters,
aDd (ii), the actual modeling of the specific conditions of
interaction betqeen the medium and the body surface.
In one of the first publications [5], the form of the relations was chosen in the follcndng class
of functions:
o. =E r9(")q9, 6=p,
t
i(3.1)
This makes
it possible to present any integral force charaqeristic of the medium action on the body (component
of force) in the form:

c=E
0=p,
r
Here, for prescribed
mlues of lP, the shape
functions
g.(t depend
only on the body shape and orientation of the
body with respect to the translational rclociry and the regime coefficients tf) "t" determined, in the main, by the
conditions of interaction betqrcen
the medium and the body surface and by the surface properties.
Representation in the form (3I) turned out to be tairly conrcnient and was used later mainly for to two reasons.
Firstly, when the integral characteristics
are knmm the linear dependence
of the rigbt side of (3.2) on ljo simplifies
the solution of the problem of model parameter identification, and the first procedures
of this kind vere dercloped in
the initial stage of study [7{. Secondly, the structure of relation (31) na}es it possible
to derclop effectir,e
methods of
calculation.
In derrcloping
particular models for the intermediate region of rarefied gas flow, the follwing intuitile reasoning,
which was formulated most precisely
b 19,
n 5J aod later taken into account by most researchers, was used: (i) the
requiremeat of continuous passage
to the limiting regineg (ii) the requirement of satisfaction of the prescribed
conditions at €=0 and O=tc/1(iii) sinplicity and efEciency, the lafter being associated with a recommendation to use
3 ninimum complete systen of orthogonal functions. On the basis of these requirements, a specific system of
trigooometric ftrnctions was proposed-
A subset of models (3f) introduced in [31] with the form
Or-E/,cod0, O,=sin0E4"outO (33)
is more widely used. It includes a large number of uiell-knocm
particular cases
and presents certain adrantages
for
dereloping specific methods of calculating integral characteristics
For the intermediate region of rarefied gas flow, the problem of dercloping specific mode\ which take account
of the influence of the gasdpamic flow conditions (prinarily, the Ifuudsen and Reynolds numbers) has been of botl
theoretical and practical interest.
The simplest approach to the solution of this problem, which is still topical [61], prwides for the application of
semi-empirical
methods for aerodynamic coefEcient
determination
in the intermediate region on the basis
of the ulues
of the coefFcients in ndenseo
gas and free-molecule flops.
This approach is discussed in a review lectue [L05J,
in which the corresponding references
and the foilorring
formula are presented:
C-=|Cin + (1 - 9)C- (3.4)
Here, C-, C; Ci are the body drag coefficients in the intermediate regime, in free-molecule flow (Ifu - -;,
and in a "dense"
gas (Ifu - 0), rcspectircly; F is an empirical coefFcient
depending
on the I(nudsEn number Kn. In [102],
one can find an analpis of particular dependences
in the form (3.a); in [109], the corresponding approach is derrcloped
for both force and thermal characteristics,
and it is suggested
that an interpolation of the "limiting regimes"
be used to
calculate
the pressure
distribution orcr the body surface.
This approach prwided the framework for a series of studies I\ 5, L1\ 77, 59, ffi,79,80, 81, 8?. LL2l in which an
expression for p in (3.a) was specified
and its dependence on the flw parameters
w.ts studied using experimental data.
In [1q, a specific form of the firnction P(Kn) was proposed
and the unirersality of this function for tlpical bodies (a
sphere, a cylinder, a cone) was established. In [2], on the basis of data processing
for sharp circular cones other
par:ameters
of the firnction FG<n)
vere obtained-
In [5q 60], the foloning formulas vcr€ suggested
and justified:
F
=s(A), A
=r,ahu;
Here, Rei is the Repolds number, tn=T-fTs is a temperature factor, mdT. and fo are the body surface and
st'gnation temperatules, respectirclg S is a function obtained.from experimentd data processing.
It was found that the
choice of a characteristic length is an essential factor influencing &e accuracy of these models [60, 80].
In local interaction theory, the signifrcance of the models (3.3) an4 particularly, the non-dependence
of the
parameter p oa'the body shape
(established
for a certain set
of configurations)
goes
beyond the possibility of calculating
the body drag coefficients for the intermediate regime of rarefied gas flor. If the model form is knovm, for example,
for linear models (3I) based on equalities (3.2), &e ralues of Q- calculated
in accordance with (3.4) can play the role
of experimental
data for identifying the model parerneters
[2]. Other methods
[S, S], which make it possible
to express
the coefiEcients
for the intermediate regon analytically in terms of those for the limiting regimes, harre
also been
suggested. As a result, there appears
the possibility of calculating not only the body drag coefEcient
but also other force
y2
E tttglo
t(3.2)

and torque components.
In theoretical and experimental
studies
16241similarity paremeters
nrre found and justified. These include: the
Mach number of the free stream M; the Repolds number Reg
with the viscosity coefficient p calculated at the
stagnation temperature To,
the exponent
n in the temperatue dependence
of the viscosity,
the ratio of specific heats y,
and the temperature factor r,. For large M- the most important parameters
are Rq, f*, and y, For the hypersonic
flov
regime, the model (3.3) [54], which includes the functional dependencies
of'the regime coefficients on the par:ameters
Reo, f- and y, turned out to be fairly successful.
Further derelopment
of the method of [54], applied mainly to slender
bodies,
was carried out in [55, 72), nwhich additional parameters, taking into account
the longitudind and transrcrsal
dirnensions
of the aircraft, and M- vere incorporated in the model coefficients.
Somewhat
differenl as conpared with
[54,55j, approximation formulas were obtained in Fl.Other local and quasilocal
models and methods,
which take into
acconnt
the finiteness of M- surface
curlature [?61,
and surface
roughness
[9[], hae also been suggested;
rariants for
slender
bodies
vere derreloped bIn24,8L1.
Methods of model identification using
the elperimentally determined
integral characteristics of bodies aod a plate
vere perftcted in F8, 81, 951. Fairly general methods of inverse problem solution for bodies of rerolution and a
corresponding
review are presert€d in [801.
Since
the local models
suggested
for the intermediate
region of rarefied gas
flow are semi-empirical the wrification
of their adequacy using experimental data is rtry important. To a greater or lesser
ocent, this problem has been dealt
with in most papers on the topic under consideration
and it has also been the subject of specific sndies [41.
In the limiting case
of free-molecule
flo*,, the local character of the flow interaction with the body surface
has been
demonstrated
rather uall theoretically,
but uncertainty in the choice of certain parameters
of the model leaves room
for fiuther development.
Among &e rarious approaches
to this problem (for emmple, [10U), as a means
of improving
model aocuracy,
nr mention taking into accormt
the e-dependence of the coefficients characterizing the momentum
transfer to the body surface
[28,82, 111].
The model used for calculating the heat transfer to the body surface in free-molecule flor belongs in thE local
model category. In the case
of ndensen
gas
flw, the existence
of local models for heat flru calculation has formed the
basis of a series of studies IL6, nl using a general representation of the heat fltx as a frrnction of the angle e. In the
papers cited references,to
particular dependences
are giren. In [14] and later studies, an analysis of which is giren in
review [18], the radiation heat flrucs to the surfaces
of three-rlirnensional
bodies and bodies of rerolution in an air flow
with rclocitie.s
ranging from 8 to 18 kn/s at heights of 40{0 km qiere
studied for a wide range of body di-ensions.
The result was a formula for calculating the local heat fluxe.s
in spatial and a:rislmnetric flons'whid in the case of
bodies of rerolution, generates
a local nodel. A conectiw heat flrx localization property has been detected
with tairly
high accuracy
for certain families qf thrce-.limensional
bodies, for exanple, for elliptic paraboloids in hypersonic
flow
t661.
Hence, heat flux localization @curs in free-nolecule flw and in some cires of odensen
gas flov. Thus, in the
intermediate region of rarefied gas
flow there are identical rearions
for assuming
the local character of both thermal and
force actions.
This has made it possible to ertend the methodologt dereloped earlier for momentum transport to the
process
of enerry transport in the intermediate regime [4, U0].
4. CALCUI,AIION AI{D OPTIMIZITIION OF INTEGRAL CIIARACTERISTICS
ON THE BASIS OF PARTICT.]I,AR LOCAL MODEIS
If a model of the interaction betqcen a medium and a body surface,
w[ich makes it possible
to determine the local
characteristics
of the force and thermal action at ary point on the surface,
is knourq then the total characteristics
can
be calculated directly by integrating the corresponding
local characteristic
qer the body surface. In the case of the
particular r"odel (31), the calculation of the shape
functions for typical bodies can be simplified by tabulating these
frmctions [9] or by s$sv 6sans [35, 801.
Problems appear when bodies of complex configuration (in particular, orbital spacecraft
with orposed solar cell
arra),s)
are studied- This is connected
with difficutties in determining the illuminated surface.
Ewn in the case
of the
simplest local models (boring the action of reflected molecules), the need to reduce the calculation time requires
the
derelopmeirt of 'specific methods and algorithms,
particularly when rariations in the body orientation in the flow must
be rapidly taken into account [6,68].
These
problems
are geaeral and the efficiency
of the methods derarloped
does not
depend on the choice of a particular model.
The method of "differential equations"
is a specific calculation
method of local interaction theory. Orrcr the entire
range of angles determining the bodt's orientation with respect to a characteristic motion of the medium, this method
reduces the problem oftotal characteristic
calculation
to the solution of a recurrent system
of differential equations of
either the Poisson
(for three-dimensional
bodies) or Legendre (for bodies of rerolution) t1pe. Since a general solution
v3

form for these equations
is knoum,
the problem reduces
to the determination of particular solutions and constants. As
compared
with orrdinary
integration qer the surface,
this method! efficiency is attributable to the possibility of saving
time and elleining approdmate solutions in series
form. It is possible
to distinguish
the follorving-two main stages
ii
the erolution sf this method.
The first stage,
represented
by [107, 113, 114]
corcrs the period fron 1!)65
to tgT3when the method was dereloped
using the simplest Nefion model. The second
stage
began
with passage
to the class
of models (3.3) in connection with
the generalization of the method. The fuadanentals of this generalized
approach
nrcre
deraloped in [3! 32, 45, tl1l;
later publications
([?7' ry,/21, atd others) ucre deloted to the dJrcbpment of its rarious aspects,'methods,
algorithms and associated
computer programs,
etc. By the end of ttre sercnties,
the deralopment of the theoretical and
practical aspects
of the method for the class
of models (33) had made it possible to include it among the standard
methods of integral characteristic
calculation.
The further derelopment of the method was
associated with &e use in (3-1)
of functions more general in form than
(33); in the monograph [80], a corresponding
approach
and references
are presented
in detail.
Practical needs naturally gare rise to the problem of determining thJ aerod)'namic
characteristics
of bodies in
aontranslational
motion. The use of local models
for these
purposes
had a long history in connection
with the calculation
9f t!! fgrcel and torques in free-molecule
flow [53, 70], for c/hiO as a rule, a sinptiheA model [116] was used.
A more
detailed
aaalysis
of aerodynamic
characteristic
behavior was
caried out by calculating
and studying rolational deriratires.
4-log the papers
based
on local models,
urc note [106] (Newton nodel), a series
ol studies (n, ag,98J (free-molecule
flow), and also a study [3] in which analytical formulas fss similar conditions ucre confirm"a "u-iri*Ury.
- - O|e of the complicated p.roblems
that arise in calculating
nonstationary aerodynarnic
characteristics
on the basis
of local models is that connected
with the dependence
of the fuuminated surface
boundaries on the angular rclocity of
the body' which is often ignored in order to sinplify the calculations.
A study of this problem is one oi th" i-pott-t
features
of [98] rvhich demonstrates
thaq within the hyperthermal
apprwination [116], the rotation does not afiect the
structurc of the illuminated body surface.
In the middle of the sereutieg in the field under consideration
studies of the intermediate region of rarefied gas
flon-began.
!" [S4' numerical rcsults for rotational deriratiws obtained for certain sets
of bodies are-presented.
Further
ltodiT [a5' 56t 901 urcre
-performed
usi"g analytical methods
on the basis
of the model of [54, SsJ.
Ti"y resulted in the
fornulation of relations for calculating
the aerodynamic
characteristics
and rotational deriratires of O" ntrt and second
order and made it possible to detect the specific features of the body rotation effect on the force and torque
components.
The model of Wl was desiped for numerical solutions with body rotation taken into account.
So far' there hara-\en.a$o9t no-experimental
resuls or "exact'calculations
which could mafte
it possible
to
nalyze the adequacy
of theoretical results obtained for nonstationary
motion of a body in the intermeaiate region of
rarefied gas flow but based
on models tested for steady motion.
_ An analogous
situation arises in the srudies of body s.hape optimization in the intermediate region of rarefied gas
flotry
which begao at the end of the seranties [3?, 42, u,Ce, S5J.
Nercrtheless,
the results obtainedfor bluff bodies of
revolution may be considered
fairly accurate
because
the basic model t54l has been cell tested.
For free-molecule !w and hypersonic
flow of a "dense"
gas,
rariational problems of body shape
optimization on
the basis of local models harrc
been studied mainly within thl framevork of the approaches
of tZ4 ff3l, and some
researchers
(see
for exanple, [29]) hare retumed to the problems
of [118] with the ain of studying them dmore detail
and formulating them more correctly.
rec€nt )€ars' more and more attention has been paid to the study of body shape optimization problems for
bodies moving in a'dense'ga$ with the heat transfer to thl surface trcated as a critlrion or limitation. A ieview of the
corresponding
studiEs may be-found
in [1S].
These problems
vere also
studied in later publications [19, 54. To calculate
the pressure
on the body nufacc, the "local'f.IEwton model was used,
and to calculaie &e heat g*t, Uo6.toot "oa
more complex models.
Th-e
study [19], in which the model of [1aJ
with account for narea
laurls'
[16] was used
to calculate
the heat flux to a three-dimensional
body surface,
c"" be considered
as an exanple of the us! of a Uca model. Body
shape-optimization problems are essentially
multicriterial and the traditional way of taking this into account is to
formulate a target functional for the main criteriog with the others ta&en into account in terms of limitations. The
approach
of [5{, which takes
account
of the multicriterial aspect
in the initial formelization of the problem,is best suited
to the situation.
Prrogress
with model design and the app€arance
of effectire calculation methods prwided the basis for the
dertlopment of.corresponding
computer programs
(l\n,48,719..f.,1121and others), and the eighti;opie thaperiod
wlen wide practical use
of 'local" methods
began.
This was rcry importani both for applications
--d fo, the derelopment
of +" -ft"9ty itself since the feedback w:rs a source of uasll-founded
estimates for-the analysis of local modeL asd
metlods, ffius stimuhting the deralopment of the theory in some most important directions.
w

5. STT]DIES OF TIIE GENERAL PROPERTIES
OF I.OCAL MODEI^S AND TIIEIR APPLICANONS
Since the end of the serenties,
the development
of local interaction theory has been characterized
by nanrally
increasing
interest in the study of the most geoeial properties of local models,
resulting from the specific representation
(U.) rdthir than from a partitdar model discribing thi action of a medium on a body under certain conditions. Local
ioteractioo theory is often used
in circumstances
*hete there is no traditional theoretical
justffication of the model form,
so that the results
folloring from the assumption
of "localization"
may be more rpliable than those
based
on a particular
model for which there is no convincing
justification.
There is a certain analog5l betweeo
ii-itrtity theory methods
I9q and the studies
which resulted
in the derelopment
of methods of calculating th'e total characteristics
of the action of a medium on a body moving in it without requiring
information about a .p.Jfi" interaction model. In essence,
the local interaction hpothesis postulates
local similarity of
action on small areas
identically oriented in the flon' and interacting with the medium under "identical conditions".
The prerequisites for the deralopment of calculation methods using inrariant relations betciEen the integrd
characteristics
o1 bodi"r uiere laid donm in [36, 1081,
in which a technique for formulating linear relations based on a
knowledge
of particular models was worked out. In later studies
[13, 65, 1()3], fundamentally new results which-
made
it possiUte to iormulate unirprsal (inrarian$ relations that remain %Iid with rariation of the intetaction model (with
consenation of the generul p.p".ty of localization); for bodies
of rerolution and of some
of more complex
shape,
these
relations also remain raru witn-rariition of the angle
of attack (identicat for all bodies). The Existence
of such
relations
makes
it possible to calculate
the integral charactJristics
of a body by'expansion' ocr th9 basic bodies whose
integral
characteristics
ar€ knom, i.e. the -6Ooa uorks ercn when the particular model needed for dirEct calculations is
nissing. The ideas enbodied in this method uiere dereloped
io t12l in connection
wi& the formulation of appruimation
relations.
The generrlized approach also turned out to be useful for studying static stability. For certain bodies of simple
shape
(a .i'ght "it"ul- 6o", u redge, a plate), the fact that the pryssune
center position is independent
of the angle of
attacf was prowd on the basis
of tte NJwtooor mor€ exact
models
in t761.
Many putlications ([67193,-84land gthers)
resulting fror " long series of studies of star-shaped
bodies are directly or indirectly comected with the inrestigation
of stabif,ry
problems on the basis
of the Newton model. In this r€spect,
the study [83] is interesting because
some
of its
results are based on the asslnption of conical flov on the body surface rather than on a specific model Within the
framework of this general noael it was shorm that for sharp elliptic cones the pressure center position is independent
of &e attack angle and of the mnditions of flow action normally to the body surface.
Fgrthe6tidi"r t3d, I03l demonstrated
that, ir addition to ciones,
this property is 1alid for the follwing classes
of
bodies
with elliptic cross
sectibns:
segments,
cylinders,
combined
bodias
consisting
of cylindrical and conic elements
with
a flat or (for appropriate geometril dimensious) ellipsoidal nose (ia particular, segment-cone
and segnent-cylinder
bodieg double cooes,
etc.):This property is also-ralid for a wide class
of conic bodies whose cross section is bounded
by segments
of straight lines or by-arci of an ellipse, in particular, for right pyramids and diamond-shaped
wings.
Iriathimatical means for the geometrical design
of such
bodies hare been dereloped.
For a wide range of fl& conditions -d ratious configrnations,
narEa
lau/s"
are knocm, according to which the
integral characteristil of different bodies sre s,irnilar
if the rariations of their cross-sectio"al
areas
in the flow direction
are identical and certain additional conditions are satisfied.
A review is giren i" 1151.
In a number of papers, loul or
simits' approaches
hac been used to justify tle narea
laws".
On the basis of the Newton model in F6l the "area
laws'
*ere derircd for the drag while for the heit fltrxes and the wake behind a three-di-ensional body research
based on
the localization concept I presented in [15, 16] and ffi, respectirrely.lnl42l, the validity of the 'area la*'s' for the drag
on a body similar to " Uoay of rerclution ana of the analogues
of those lav/s
for certain other spatial confrgurations
was
established
for a general class of local models.
' A consideraiion of body shape
optimization problems on the basis of the general local model of [40] resulted in
the determination of aerodlmamical
forms whose optimal properties
do not rary with the particular local model.
An inportant stage in the devElopment
of thi general iocal interaction theory was its extensioa to the case of
nonstationaiy motion pe1. Whereas
in ihe translational motion of a body the ralocity of each
point on its surface
is the
sr-e, and tbe influence of tUs relocity on the integral characteristics
can be directly or indirectly taken into account
in terms of the global palerneters
of the interaction betqpen the body and the mediuo, the presence
of rotation makes
it fundanentally necessary
to take into account the rclocity distribution oler the body surface.
In general taking this
factor into account t"rults in the need to include the local relocity in the nodel as a second local paremeter.
Newrtheless,
the large number of knoum
local models
makes
it possible to keep functions of only one global parameter
when exending thelescription to the case
of rotating bodies. Iater studies [3, 39, 4L] were carried out for a fairly
general subsetif local moiels. For bodies of rerolution at lov angular rclocities, formulas for calculating thc fir,st and
Iecond-order rotational deriratiws upre obtained.
As a result, certain patterns in the integral characteristic behavior
v5

detected earlier for particular models could be generalized.
In connection with the discussion
of the general
methods
of local interaction theory,
we should mention monograph
[80] in which a methodologically uniform approach
is presented.
The relation betr',een the local theory and the problem
of Markw monents indicated in [80] turned out to be rery fruitful" in particular, by making it possible to use the
corresponding mathematical apparatus to sohe a nunber of problems.
6. NONTRADMONAL AREAS OF APPLICAIION OF LOCAL INIERACTION TIIEORY
Supersonic aerodlnanics, gasdynamics,
and the modeling of luminous flux action on a body can be coasidered
traditional areas
of application of local methods.
At the sane tine, in other fields of mechanics
and for other conditions
of interaction betqeen the medium and the body surface correslnnding approaches harire
also been dereloped.
In [88], an attempt was made to apply the locd approach to subsonic
flov past bodies. In [75, 93], a local model
for describing the action ef 3 highly rarefied plasma
on a body surface wati proposed
and total force calculation formulas
for bodies of simple shape nrcre
derired. tn [94, on the basii of the approach
of [a3] to the modeling of free-molecule
and similar'floqn in cascades
and channels and on permeable
surfaces,
a local model for &e description of hypersonic
free-molecule
flow action on a mesh-t1pe
surface
was suggested.
In a number of papers [30, 50, 85J,
local interaction models hare been used to describe the action of soils and
metals on a body surface in connection with problems of body shape optinization fss high relocity motion in or
penetration into these media. Certain prerequisites
of this approach
c,ere defined in t30, Sa Ag]. iheoretical and
experimental
studies indicated that the Newton model giras impactor shapes which nake it possible
to penetrate deeply
into soils and metals. The problems of shape optimization for a penetrating body differ fundamentally from those for
bodies moving in a medium, since one criterion is nov a certain integral characteristic (usually,
the penetration depth)
determined as a result of integrating the body motion equation formulated with account for the rariation of the contact
surface
during the pnetration process
[891.
In a number of complex problems, local models are used as the initial appruimation of the solution represented
in the form of a series expansion
in a small parameter characterizing
the problem under consideration. For orarnple,
in studying
the thrpE-dimensional
hypersonic gas
florvs past
wings
by the urcll-knoun asymptotic
thin shock
lalcr method,
the solution is sought in the form of an expansion
in a small paraneter equal to the density ratio at the strong head
shock
[58], with the Newton flw scheme
taken as the zero approximation,
corresponding
to an infinitely thin so6pr€sssd
lapr.Thus, approaches
based on local interaction theory hare turned out to be fairly fruitful in rarious fields of
mechanics,
with the range of applications expanding as the theory has deraloped. This naf,es it possible to predict
ialsresting new results.
REFERENCES
1. Yu. V Abramonich and E' P. Shiroltoplps, lAn cngincering mcthod of computcr calculation of airuaft acrodynamic charactcristics for
hylrrsonic vrlocities," Ir TIAGI, l4 3 Qnq.
2. M. G. Abramock4a and V P, Bass, iA, study of the acrodpamic characteristics
of cirrtlar concs ir the intcrmediate flow regim e,, IJch ?ap,
TSAGI,
l\ NoJ, 122 (1980).
3. M. G. Abramovska:a and V P. Bass, iAcrodlDanic charactedstics
of roatiag bo<lics
in a nrulticompooelt flon' of rarefied gas,'
Izu. Alcad.
/\ba& SSS4, M*JL A&k Gaza, lrlo.6, 166 (1981).
4. M. G. Abramovska5a
and V P. Basg 'Ol thc calculation of thc aerodlmamic characteristics
of bodics in the intermediatc flow regime,' in:
Space Sadies h Uleabe [in Rlssianl, VolI{ l.Iauloa Dumta Kicr' (1982), p.29.
5. M. G. Abramowka:a, V P. Bass, A" V. Limanskii, and V I. Timchcnb, 'On the calculation of the acrcdynamic charactcdsti6 of bodies
in thc intcrmcdiate lloqr rcgimc,' m: Applicd Aaodynanics of Spacectaf [in Russiaal,
].{aukoa Dumka, R* (|fi, p-69.
6. M. G. Abramonka5a, V P. Bass, V D. Pcrminor, and .d V. Shrador, 'On cefiain dercloped algoritbms for calculating thc aerodynamic
charactcristics of airraft in frrc-molcculc flcln' Tr.Ts,&I, A%, 6 (lffi).
7. O..4- Akscnoa, "Thc dcpcadcncc of thc rcginc cocffrcictrts on thc rcgimc parameters
in local intcraction theory' Vgdk LGU. SqJ,
No.1, 100
(1989).
8. E V Aleksecva and S. N, Aleksco,ra,
l{n account of frnite Mach numbcr influence on the rcgme coefficicnts in local interaction thcory'in:
Aaodynamics of Rozfted Gases
[in Russian], Voll.l, LGU Leningrad'(1983), p243.
9. E V. Alekscsrra and R G. Barantscv,
I-ocal Mettud of Aootyna,mic Cabubtion in a Poefied Gas [in Russian], l,GU, I"cningrad (196).
10. S. N. Aleksccra, R 6. Barantscv, and I. A. 6,ndeq,
'Local interaction thcory with account for thc fioite Mach number' in: iaoaynanics of
P@efred
Gares [in Russian], Vol. 8, LGU Lcningnd (1976), pX62.
11. S. l{. Alekseaa and R N. Miroshin, 'On thc dcpendence of the local interaction pafttmeter on the Knudseu numbe6,' in Aaodynanks of
Ratefied Gcses,
Vol. 7, LGU, Lcningrad (19a), pJ80.
12. A. V Antonets, i{pprocimation of complicated patameter dependences by the choice of weighting coefficients in the arrcraging
formulas,'
Iat-4Jcad"
ikue.SSS4 Melclt Mk Gaza, No. 1, 161 (1!tB6).
%

13.
15.
16.
17.
18.
19.
m.
2t
))
23.
u.
25.
26.
27.
a.
a.
30.
31.
32.
33.
35.
36.
t7.
38.
39.
4.
41.
42.
'd V' Antonets and A V Dubinskii, iA' method of calculating the aerodlmamic
characteristics
of bodies on the basis
of the invariant relations
of local intcraction thcory' Prikl. Mat MekJr, No.5, 872 (Dg3).
E Z Apsht6ia' N' v vartanlan, and V I. Satrtrarov,
"on thc distribution of thc radiatira heat flu:<es
on thc surfacls of thrcc-dimensional
aad axisymrnetric bodics in a supcrronic flow of inviscid gas,"
In Alc4d. tfuiksss4, Makft. zhidk Gazz,No. t % (1g86).
6. z npsntein and N. G. Kolotiloa, "Thc arca law for intcgral parametc$,'/2L Atc4d- t*luksssg Maka zhidk Gaza,No. 4, 157 (1991).
€' Z apsntein and N. N. Pilyrgin, The arca law for thc hcat transfer coclfrcicnt of ablating threedimcnsional bodics with a local
g"ry"::T::j :1"_1?l,nY* on
thc
surfacc
inclination
angNe,'
Iz. Akad.I\bz&
S.ssi,
MekJL
Ztdik Gaza,
No.
z,7r (rs79).
E /- Ap6ntetn and N. N. Pilyugt rsteadrstate shape
of bodies destro)Ed
by hcat as a function of thc surface inclination anSe,,
Izw Alcad_
/ez& SSS4,
Mekh ZNtIk C,,azo,
I$o. 6, 137 (1991).
$ z afe-lt9in, N. N Pilyugin, V G. Sarastlancnko, and G. A. Tirkii, 'Radiatira hcat transfer of bodics cnrcring the atmo6phcres
of thc
Earth and plancts with extraorbital rElocitieq' in: Itogi Ikuki i ru*nu Mektl z,titk c"r- ti R J;i;voi ;,-vnml, ulsco*, 1r*g,
pJ16.
M' A" Arguchintscrra and N N- Pilyugin, 'sPatial shapes of bodies with minimum surface hcating in hypersonic motion though the
atmoslrhcrc,r Kosttt Islcd, |Io.5, 615 (1992).
E N' Bondarw V T Dubor', Yu .A- Rphor et al. (cditors), AooWoncclanis [io Russian],
Mashinostroenie, Moscor, (lg3).
4ryY-nbs of Rozftccl Gaes [ia Ruscian], Vol. 1_11,
LGe Lcningnd (1%]19S3).
R G. Bannserr, htqaction of Poefut Gascs with Ftant
Swfaca ldnussianl, l.lauka Mcccnr (1g5).
RG'Barantset/, 1{locatthcoryof momcntutnardcocrEf traDsfcrtoabodyiurfaceinarat rr.d gaqaia:MattsnaicatModzlingAnatytical
and Nwtqical Malbdt in Twtspon Ttwry lln Russianl, Minsk (1@), p.90.
R G. Barantscrr,
1a uriant of thc local mcthod for slcnder Hif ! ; rat n o ga4. vxt LGIL Sal,vol3, Irb. 13, 99 (19g2).
R G. Barantserrt
L .d Vasil'ev, E. V hanorr, V. V. tr(ozachclq
-d D. Minaichcr', f V fr,fitb"if-, and N. V Murzw, 'lAcrodpamic calculation
bascd on the hlaothcsis of tocalization in a rarcficd 9?N.:'
ia: Aadyutnics of Ro@ Gasa [i! RNsian], vol. 4, LGU, Leningrad (1g69),
pJ70.
R G' Barantsct' and V- M. Fcdotora, 'Nonisotropic effcct of ornaturc in local intcraction thcory,' ia: Aaodtrnotrics of Rozfutt Gasas
[io
Russiaal, Vol. lQ I3U, Irningnd (1980), pl%.
v P' Bass and v I' Tinoshcnlo, iA,pplication of thc local btcraction mcthod to calculation of the acrcdldamic charactcristi* for bodics
9! Tgpt"* shape in a hypersonic
flow of rarefied gas,. ?t TIACI, t&A $ym.
V V Bclctskii ad A" M- Yawbn, Aaodynamic Force
Efeu on tle Roating Motiott of Atnfuiot SauUins [in Rnssiatr],
I.{auhqa Dumka, Kicv
(1!n4).
V L Berdichcrnkii,
'On a_minim{
ong ryq shape
in a lqipersooic
gas
flor,' Vst MG(I SqJ
Ma-Ma&fr.,
I\tro.
g,n Gns).
V' S' Bondarchuk, Yu. .d Vcderni&or,
V G. Dutor, and V E Minin, iOn the optimization of staFshapcd
impactors,. &y. So lrV SSSR fer
Tbkll l&uk,3, I{o. 13, 60 (19S2).
'd I' Bunimordch, 'Rclations bct*ten the furtcs acting o1 bodics modng in a gag in a luninors flux, and io a lrypcrsonic ]Ie*toniaa flonr,.
Izu.
Akadl\&..& SSSB MekJL
?J&tk C,424,
NIo.
4, g (LgT3).
f, ! ni$molch, l{crodynamic characteristics
of axigmrmctric bodies in a flcn, under'localization la/ conditions,. Vqm MG(J. SqJ Mat
Me.kh,lft.4,n (ln4).
'd I' Bunimordch' 'The "localizatign lau/ in aerocllnamics
and rarcfied ga<
dynamics,' ia:
pnc, 4 So9.
Conf, Dyrr. R@et Gasq and Molec. Gas
Qn;, !'tolcortt 1975
[ia Russiaal, TsAGI, Moscor, (l9Tl), p39.9.
'd I' Buninovich, "Iel inter:action
thcory in rarcficd gas dylamicq' ifi Aaotla t dyrrarnics in Rozfied Ftass [in Russian], l\dAI, Moecw
(1988),
p.25.
A I' Bunimodcb, M' .d vototyltserr, and N. I. Sazonoa, 'On thc problcm of dctcrarining thc aerodlmamic
characteristics
of bodies iomcrsed
in a uniform flow in "localizatioo lac/ conditions,' ve*t" MGU. snt uot Mekh,th.e, rrr 1rrsj.
'd I' Bunimot'ich and A V Dubinskii, "Generalized
similarity larc in tto*s past bodics undcr ilocaiization laq/ conditions
,. hild, MaL MekJL,
I{o.5, E57
(1973).
'd L Bunimogich and A V _Dubinskii,
'Optimal bluff bodics of rsolution in a rarefred gas
with rariablc density,
" I*. Akad- IvottkSSSB
Mak r
Zhi<k Gaza,I{o. 3, 158 (1980).
'd I' Bunimodch aad,{. V _Dutinstii 'on thc pFssulE ccotcr of bodics,' Izv,
Akad.Ivz*.tssR Mekh zrfurh 6aza, NIo.
s, LD (lg&).
'd I' Bunimorich and A V Dubinskii, lAcrodlmasric
charactcristics
of a$itrarily nating bodics io a ra:eficd gas
of rariablc dcosity,.
Ibmt.
I$led, tl, Ilo.a 1&) (1989).
'd I' Bunimcpicb and A V Dubinskii, "Bodies of minimal drag in a rarpficd gas,' in: wory hobbts in the Mecta nics
of D@nabtc Media.
ft.I [in Russian], MCU, Moscow (1F0), pJlg.
'd I' Bunimonich and A V Dubinskii, 'On thc calculation of thc rctationat dcd%tirts in the 'local' intcmction of a ftowwith a body surfacc,,
hild MaL Mckt,$,I{oJ, 56 (1992).
- 'd I' Bunimcpich, A' v Dubitrsldi, aod V I. Kru'mcnko,'!n thc aerodynanic and thcrmat charactcristics
and optinal shapc of bodics in a
taTfi{ F flo*,'in: Rozftcd Gas
Dyanic* Pre.6 so,,.
Conf, Pn M{,dbtskfl2 [in Russiaa],
Inst. Teplofiziti, t*orcciuirsr (1980),
p.49.
'd I' Bunimoich and M. L tGgan, 'Frce-molerulc gas flw in channels
and gids,' IA'. Alcad. ibze sssi, Ivr*tL ?nak Gaza,t&.3, !29
(1%6).
A I' Bunimorich and v I. Kuz'mcnb, lAcrodyriamic and thcrmat charactcristics
of threedimensional star-shapcd
bo<tics
in a rarefied gas,'
In AI@d. /ka& SSSB Melctt Zhidk Gaza,I,to.4, nf (1993).
A" I' Bunimodch and N. I. Y^oT', f.l*lyo?t T:thq for dcrcrmiaing the actodlmamic
forces
and torques in rhe nonstationarymorion
of lodics th$ugh a ncfied gas of ririautc density,' i$ Gas and woe Dytuttdts [in -fussianl, vol. 2, MGfi Moscorr (L979), p3L
A I' Bunimorich and V G. Chistolinor, iAaalpical calctlatioo of thc aerodlmamic
charaderisti6 of bodies of F\Dlution under localization
faw cooditions,' I^2 Akad- j\br* SSSB Me*JL ZJridk Gaza, lir.. S, % (lLlni).
'd I' Bunimodch and V G. Oristotinov, 1{n analytical method for calculating
thc aerodlnamic forcs in a threcdimcnsional problem under
'localization
lac/ conditions,,
hild Mat M*h,9, No. 3,466 (lg5).
'd I' Buoimodch and V G. Chistolinov, ltn analpical method for determining the aerodynamic
charactedstics
of bodies in a hypersonic
flow
of rareficd gas
with rariable density,' Tn Ts,4GI,
l&tq 11 (lyfD.
45.
6.
47.
€.
v7

49. A. L Bunimovich and G. E Yakunina, 1{ study of the cross-scctional shape
of a conic thrccdimensional body with minimum drag moving
in a rarcfied gas,'
Im. AJIaA I&uk SSSB MeA Zhitlh Gaza,I{o.5, 112 (1!)86).
50. .d I. Bunimorrich and G. E. Yakunina lOn thc shape of bodies of rerolution with mininral dra& not/ing in compressible plastic and
etastoplastic media,' hild MaL Mekh,Sl" f{o. 3,4% (1987).
51. L A. Yasil'ev,
Detomination of bght Prarure on Spacecmft
[in Russianl, Mashinoctrccnie, Moscow (1985).
52. F, E Vitman and V .4* Stepaoor', Thc effcct of the strain firtc on thc rcsistance to dcformation of netats at impact rclocities of fd-fd
m/s,' in: Sonc Problems of tle Sazlllgth of Solids [in Russian],
AN SSSR,
Moscur, lcningrad (1959).
53. V S. Galkin, 'Dctcrrrination of thc torqucs and forccs acting on rctating bodies in fiec-moleculc flcnr and a luminous flu*' Inzh 7.lue5,
No.5,954 (1965).
54. V S. Galkin, .{- I. Erofccrr, and .d I. Tolstyth, iAn appnximate mcthod for calculating tbe aerodynamic chaactcristics of bodics in a
hpersonic 0cn' of rarcficd gaE,'
Tt Tl4Gl No. 1833,6 (lym.
55. V. S. Gatkin, .4" I. Erofeo,, and .d I. Tolstykh, 'On an apprcximatc mcthod of acrcdlnamic calculation in a rarefied gas,"
?i Ts.€/, No. 2111,
n $eBr).
56. V S. Galkin and L L Zrorykin,'Rotational derivatires of bodics in a trypersonic rarcIicd gas flow,'Tn Ts'4G1, No.2220, 3 (1984).
57. O. ,{. Gil'man and N. N. Pilyrgin, 'Paato optimal shapes of axiqmmctric bodies at higl supersonic
\Elocities," Prild" MaL Mek]L, 55,l'{o.. 2n
2e0
(1e1).
58. V N. Golubkin, 'Thrccdimeosionat $pcrsonic gas fl€n'past wings,' /ar, Ros. Akad- IUat( M*JL Zhiclk Gaza, t{o. 5, 148 (192).
59. P. I. Goreobukh, 'iA correlatira rel4tion for the drag cocfficients of bodi6 in a $Bersonic rareficd gas flfi,,' UclL ?4p TsrcI, 1ll, No. 2, I
(1e85).
60. P. I. Gorelbukh, 'On tbc appEoximatc
calctlrrtion of thc acrodlmamic
charactcristics of simplc bodics in a hyTcrsonic rarcfied gas flow,' ft
TsdCI, tr{o. 24X,8 (1990).
61. V. N. Guscrr, l{crcthcrmodylamics of higb-altitudc fight,' Ia'. fuss. Ald. t'Iat( Met Mk Gazn, No. 2, 1a2 (1993).
62. V N. Gusev, .{. I. Ercfean, T. V. Klimoa, V A. Percpukhor, V. V Rpbcr', and A.I.Tolstykh, i{ thcor€tical and erEcrimental study of a
rareficd gas hlpcrsonic flcn'past bodics ofsimph shapc,'71 ?tdGI, I{o. 1855,3 (1yt7).
63. V N. Guscrr, T. V Ktimoa, and .A- V lapin, 'lAcrodynarnic
chanctcristics of bodics in thc intcrmcdirrtc region at hlpersonic flosr \Glocitics,'
Tl TSAGI, NIo. 1411, 3 QY|D.
il. V N. Gusct4 M. N. Kog3n, V..d Perepukhor,
'On the similarity and vadation of thc acrodlmaoic charactcristics itr thc intcrmcdiatc regon
at tqpersmic flcn' velocities,' Udl Zap. TsAGI, t" No. 1, Z (190).
65. d V Dubinskii, 'Rclations bctvrcen thc fores actiog on bodiqs of differcnt shapc modag in a gas,'
hild Mat M&h,4{, No. t, 1?8 (1%0).
6. V V Znamenskii and .{ V arbarev, '(ilculation of thc cotrtrctirc heat flurcs bascd on a given ptssurc distribution in thrccdimcnsional
fl6r,' I^). Alcad"
teuk SSSR, Mekh" Zlndk Gaac, l.{o. 3, 160 (UeD.
67. M. .d Zubin and N. .{ Ostapcnfto, iAcrodynamic charactcristics and satic sability rrscrrc of conical sar+hapcd bodics at hypcrsonic
',elocitics,'Inr Ros. Alud lt&ltlr, Mel& ZhUh Gua,l.lo. 6, 142 (199).
68. E" Yu. Zucva, M. M. I(omaru', and V. V Sazonor, iA, represcnation of the acrodyDamic torquc in problems of mathcmatical modcling of
the rotatiooal motion of artiticial carth satcllites," I(osm Issled,I, tlo. 6 Tn Gry4.
69. S. G. Ivaaor and A. M. Yanshia, "Foccs and torques acting on bodies rotating about a qrmnetry axis in fiee-molccule flan' I*. Akad. IW
.SSS4,.Uek
. ZNdk Gazo,
No.3, 151
(1980).
70. V M. I(ovtuncnko, V E tkmcnko, anO
E" P. \*kanich,Aemdynambs of Orbiul Sprceoafi [in Russian], I.,iaukoa Dumka, Kjat (l9Tl),
71. lrt" N. Ibgan, Rar@d Gas Dyunic.s [in Russianl, Ir{auka, Mcco'(1967).
72. V M. I(otcr, E N. Lphkin, .4- G. Rcshetin, and .4. N. Shchelkooogw,
"Calculation of thc aerodldamic characteristics of bodics of conplicatcd
shapc in thc intermcdiatc region," iaz Nwwical Modcllingh AaoQAodyanfus [in Rlssian], l.{auka, Mosconr
(1!}86),
p115.
73. Yu. A. Ibshmarpv and Yu..d Rlzhor, App&ed Roefud Gas
Dytottics [in Russiant, Mashincfiocnie, Mcccn'(1977).
74. A" N. Kraiko, Vqiatiotul Ptobbtts b Gaxlyanics [in Russianl, I{auka, Mos@"'(1fl9.
75. .d P. Kuryshev, B. V. Filippor', and V P. Yartsor, 'Effect of ion thermal relocitics oo the aerodynamic characteristics
of bodies ia a higbly
rarcficd plasma,' in: Aaodynantics q RCIefud Gases
[in Russian],
Vol. 4, LGU, I*ningrad (1%9), p149.
76. l{Y.Lunet4 lllposonic Aaodynonics [in Russianl, Mashinostrocnie,
Moscow (195).
Tl. R N. Mirchia, 'i{ linear regrassioo
analysis
of cnpcrimcnts in a rarpficd gas,'
it Aaodytanics of Roefwl Gases
[in Russian],
Vol. 5, LGU'
kningrad (190), pf4.
78. R N. Mfushitr, "Lincar indqrndcnce of thc shapc fuactions of sharp cirrular concs in local intcraction theory,' Vmt LGU. SqJ Ma MeJch.,
t{o. 2, 69 (1986).
79. R N. Mitoshin, 'On thc dqrndencc of thc drag cocfficient of conrcx bodics in a rarcficd gas on the Reynolds numbeq,"
Ven I.GU. SqJ
Mat Melch, NIo. 2, 48 (1989).
80. R N. Mitoshi! and I. .{. Kl:r.lldort, LeaI Intemrtior Thcuy lin Russian},
LGU, Lcningnd (191).
E1. -V. S. llaritsa, '!An aaatpb of qpruimations for thc acrod:Eanic coefficicnts of slcndcr bodies ia a rareficd gaS' Vmt. LGU. Ser,.
L l,{at-
Metclr,, l{o.' 2, 114 (1987).
8L .d I. Ornclik, 'Measuring thc coefficicnts of momcntum transport to thc sutfaccs of various structurl in hypersonic frc-moleclle floIn' Iz't
Al@d l&ttk SSSB M€ktr. Zhitlk Goa, tlo.4, 176 (1yl7).
83. N. ,q- Ostapcnto, 'On thc prcssurc ocnter of conical bodics,' I^'. flcaA Aeu& SS54, MeklL Zlridk Gcaa, I',tro. 1, I (1980).
U. N .d Ostapcnko, 'C-onical bodics with a star-shapcd
closs
scction posscssing a rcseree
of static stability,' Izu. Aladi\ell& SSSB Mekh TJidk
Gam,t{c.6,84 (1984).
85. N. ,{. Osapcnko aad G. E- Yakunina, 'On bodies with mininrurn drag modng in media under localization laq/ conditioas,' Izt'. Ross
Alcad-
tedg Mekll Zhi<k Gaza, No. 1,95 (1992).
36. N. N. Pilpgin, S. G. Tikhomirov, and S. Yu. Chernyarrkii, 'Thc area lan' for thc cate behind a thrcc-dimensional body," In Alcad- I&ttk
SSSB Me&tl Zhidk Gaza,IIo. 3, 1,|8 (1980).
87. V Ya. Ponomarev and V. S. $.gn, 'Calculation of tbe aerodpamic characteristics of typical bodies in stationary and nonstationary motion
thrpugb a rarefied gas,
based oo the tocal interaction b]rpothcsis,' in: Ptu. a Sou Conf. Dyn. R@efad Gass and Mola. Gas Dyn Moscory
1975 In Russianl, TSAGI, Mosccnr
(l9Tl),9.1U2.
348

88' A' E" Rizo' 'Empirical formulas for cones
in subsonic
aerodynamics,"
vesm.
LGU. serT Mat.Me&h, No. g, 116 (1%g).
P. f1. _Y1
Sagononlan, penetration [in Russian],
MCU Mosc.ow
(192a).
90' N' I' Sazonoa, 'on a problem of nonsationarv flow past uooics unier locati"ation law conditio ns,,
In A)<ad-Aeu&
.sssg Mekh zhitlk Gaza,. I{o.
3, t8 GEm.
91. L' I. Scdor, si',ikuity and Dbnmsionar Mettods h Mectunics,Acad. pcss, Ncw york (1.959).
92. V I. Timoshcnko, i{pplication of differentiat rctations f"_r
fo* anO
torques
in locat interaction theory for the morc accur:rte
detemrination
iG.Afffrffi*tT*fi,:lE;f,,T.fu:'nco gas,'
ti i&)a Gas
Dvna.mics:
proc.
6'sw.
conf.
re7e.
r%nsibi,sk
pt
2
[in
93' B' v Filippor' "Rarcticd
plasma
floa past
u&i*s,' in:
A*odynamics
of RaefudGasas
[in R'ssian],
vol. 4, LGe L,eningrad
(1%9),
p-133.
%' v D' I(habalor,
'surface rcughness
"tL.t oo the regime
coefficiens
in local
interaction
theory' vesn.
LGU. sqJ. Mu.Me&', M. l, g0
(1e8e).
95' I' I' Khofpvin' 'Detetmination of the regime cocfficients in local intcraction thcory on the basis of the data for a prate,, vesat.
LGu. serJ
MaL Mech,I{o. 2, 125
(199{t).
96' G. G. chcruyi, Gas Fraus
at Htgh supa'sonic [in Russian], F.",o,atgt%Moscow
(19.59).
", ft ;jfftail: ne:n[Hi*" oi thc
aerodvnamic
characteristia
or oii." wittr
nct*rrapca
surfaccs
in a
hlpcrsonic
rarcficd
sas
flc,w,,
:' mr":H* illi, ?rttr:, Xtffffi" or
thc
nonstationary
acrocl5namic
fones
and
torques
in
the
frcc-moreculc
rcgimc
or flon, past
99. S. G. Antonor,, M. S. Iralov, A. V Kashlo/skii, and V G. Chisolinor, ,Inflynl of_atmosphcric rarcfaction <in the aerodynamic
charactcristics
of fligbt \chiclcs,' in:
Roefud Gas
oy"**tt ni I7h i;;;rra. " noxfua Gas ayn -A;"tu4.1990, weinchei
m (l't), ps22.100' R G' Barantsev' l{n{rtical studies of ga+surface interaction
," in:
Raefici Gu Dyan&s. setectea
eqas l3th Int syntp. lbo:f,bir*, 1gg2,
Pfenum Pcss, Ntrew
yort (19S5),
p.65f
101' N' Bcllomo' & Riganti, ana M. r- Vacca,
'stochastic dynamics
of a flat darc in T:lTj": f,a,i in:
Rtefzed Gas Dyrrarrrics.
2. TeeJm
papas
12th I& sytttlt' on Raefrcd Gas
Dynanics. cttotoazs,/rh ya Be.e.ziaie.q, NIes yort (1981), pJ063.
1o2' D' E Boylan and J' L Poncq 1{erodynami6 of typical tifting bodics undcr cpnditions simulating rcry high alritudcs,' A7AA r., s,l$.2,2?6
(t96?,.
103' A- I' Bunimovich and 'd v Dubinskii,'Iocal methods-in ra1!e<l gas
dynamics,"
n: Roefud Ges
Dyanics. seteud papas r3th ltt syttp.
IfialvllniDrr*,,
I9EZ,yol.1, plenun prcss, lrlcw yort (1985),
p.431.
104' w' J' Erans' iAcrodynamic and radiation disturbance torqucs on satcflitcs haning
omplex gcometry,, /. AstonauLsci, 9, l.ro. a; % G%2).
19:. 9. Fcnari, iAerodynamic_problcrns
of rc+ntry," Meccotica, q l.io. l, A e94.D.
106' n J' Hindclang an-d-
R Meisingeg l{n*tndbarkcit dcr I'lcc/tonischen Thcorie zur Bcrechnung acro<tynamischer Beiqcrte und
Stabiliatsderivativa,. Raumfubqftrsctwng,lE, No. 2, 79 (lnq.
107. H. Jaslcnr,
iAcrodynamic rciationships intr*crcnt
in Newtoian'im pct theory,"
A|fu4.I., 6, No. 4, 608 (t1l6g).
108. H. Jaslon, "Nonaffine similarity h*, ioher.nt in Ncwtonian i.p"ct ttreoy,;'AIA/4 I,g, No. 11,
MZ (lg1).
109' r' w' Matffi lgffi;rc u;aging rclations in the transitional rcginc berc,ecn
continuum and fice motccutc florn," "t spaceoaft and
ttt' }#ilTn!fr:* ?rilrY"**r gas-florr
orcr
comcx
bodics:
a phvsico-mathematical
modcl
for hcat rransfer
catcula
fiaa,"
Appr.
Matt
111' s' v Musanor' 'd P' Nikiforov, A' I' omelilq and A. G. Frcedlendeq 'ElEerimental determination of momeotum transfer coefficient in
hlpcrsonic frec molecular flcnr and distribution function reco,ary of reficctei molcculcs,' in:
Razfud Gas
Dyamics. seleaed papas t3th Int
syttp. llowsibit*,.1gE2, plenum prcss, New yort (1gg5),
p,66g. "v'vt5.w' "1. twcJe' v6 )
L72' u' D' Perzrinor' S' L Gorelor' o' G' Freedtentl"t,
-T9 -A A. Khmelnibky, l{,ppruimate aerodynamic anarpis for compricated bodics inrarcficd gas flo*',' n: Roefen Gas
Dynanno: Ptoc..l7th InL sytttp. on noeiiea e;* ryrt-t"t.;;;"rr\--i&,vel,weincheim (1r1), ps*.
113. J. Pike, The tift and drag of axiqnamctric bodies in ncrytonian nn,, eUh i.,Z IVo. 1, lg5 (1%9).
114. J. Pike, "lrlcwtonian aerodynamic
forces from poissoab equation,,
AIAA l,ri *. O,
49 (lg$).
115. J. Pi&e,
"Forces on conrcx bodics in frcc molecula
r flo"4.
AlfuLt, lg No. ll, t4,fl (lnil).
116. D. M. Schrello, Approximate free molecute acrodynamic
characteristics,, eiS l,4No. g, 755 (1%0).
117' M' J' Hcmsch an<t
J. N. Niclscn (cditors) , Tacticai Misile Aaodynanics,AlAd Ncw york (l9g{i).
ll8. Theory
of Optimm Aaodytanic Snaps, aaa. press, Ir&w york; Iondon (1965).
v9