THE DIFFERENTIAL EQUATIONS FOR THE TURBULENT MOTION OF A COMPRESSIBLE FLUID

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• Document Page Count: 24
• Document Creation Date: December 22, 2016
• Document Release Date: April 20, 2012
• Sequence Number: 24
• Publication Date: May 29, 1952
• Content Type: REPORT

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he Dif f er~n't i~l LSua.lQr the uxl~u. n
1~ota.on of a Compressible :Fluid
Keller and 'A? Friec1n1afl (Leningrad),
L
Original Title ,
II r '.;~1=~leichun en fL.r die tuxbulente ]ewe g
plf~,crenta ~, ,
e7_ner kompressi )elen ssiakext"
a report ~d at the 1st Congress of Applied report hoax
',cs scientists. Pages 39~-LO . )
Tle~h~n~.
T,enigrad: 1921..x?
STAT
STAT
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F ON~`a FOR TURBULENT MOTION OF A CQMPRFLF FLUID
'p.-- Jflfl4Lltflm
Ill + RNTL UA .,,,.?...~
L. KFLLER AND A. FRIEDMANN, Lt;N7:IwlGISAI)
su BM IMCTE E
( ' . FRIEDMA.NN
Problem.
borne Reynolds was succe.cSful in converting the hydro-
th.e motion of a Vjscous homogenous incoan-
dynamic dynarnic equations for
av e r~ ? ? " value s of the
pre ssible fluid, so that only certain
velocity components occur in the resulting equatiafSe There
also occur 6 quantities which characterize the condition of
the e turbulence at a given place and at a given timed These
qua s rePresent 6 new unknown functions of the coordinates
n t~. tie
and the time ReynOlds' system of equations is not suff'icien't
a
to d e te rmine the se unknown quantities from their initial values.
The problem considered in this publication is the corn-
pletn.on o the characterizing quantity system of turbulence so of th
these characteristics is sufficent for the
that the knowledge of
,t
to res' ov1t~14`1
moment to find the rrc~er values of the same functions
initial
for every additional time-point. This, we wish to do by an
Sion of Reynold& ideasd The knowledge of characteristics
eaten
~d values
mentioned above is in connection with the
of the kinetic and dynamic elements for the initial moment
S We want to use the turbid;; nce in the atmosphere
For doing thy.
which :>ws.thaught to be the perfect compressible fluid.
Basis of the Reynolds Method.
The basis of the Reynolds Method iS the comparison of a
STAT
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'n tiaverage" mo,ion. For the latter
real motion W~,th a certain
of short duxata.on off' the velocity corn'
motion all fluctus,tians ?n in this
out We understand 'by f luctuati
are smoothed _ time but
uctu&tians of the quantities w1th
ease not only the f1
must be observed in transition
also space fluctuations wklich ?rb\krar
hbaring point
f rom a freely chosen point in the spaCe to a neig
in any direction. c ..
this f ashion~
oken of takes place in
The soothingaut sp
j j
'
in the place of chose l~ fl1llCtiafl 17(t C)
..---- eta value' or a certain time int.,rva1
is tak~mean
to the formula:'
J }
~
d
ing
: _ s cor
ac
) -j-
( -C---
mean value accorcU.ng to the formula:
or also aspace-time
(1)
(2)
TxY ~
tion only mean values taken according
In the following sec
be observed. The four dimensional value
to formula (2 ) will
hich the integral in (2)
~of the variable: 'tax:Y,Za over w
section.
extend5 will, be called the region 0 in the following
such iature Reynolds set
For taking the mean ..values of
tinny L. 1? Richardson (Weather
up simple rules of calcula
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6 said on the a sumptions upon which these are
predict
ss tiofS are not rigidly valid, but they
based, "a1l these a
soon as the osci11ati0n.5 are disw
show good appxoXa.ms~tions as
tr .. ~ ~ , ~a.ca.ently large number S and at random. They
~,buted in suf~
would be rigidly tuue if it were possible to so choose a
.
a~
-euala;al so that the motion could be
treated as an 'tel,small quantity and at the same time
~,nf;~na,,~
uld be infinitely large coxcaced to the periods of fluctw
wo
ua tiono '
The next step is to note that this shows that ~rithin
the interval or valu re ion used for taking mean .values they^e
. e.(
uantit1es with sufficient approxa.mation to be
are ~~~ q
hand-led a constants (Postulate I) ? From this fallow the two
main statements of Reynolds' algorithmr
(-
C
-,v1= c?Iv)
the following distributive law is validd
Further,
or limiting case of the same, we have the
and as a special
following differential formula.
() ?
a 5 as ~
where s signifies any one of the )4' basic variables t,x,y, z.
(3)
(b)
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Let us call such functiofs for which the above postulate
roxi,matiaxi with proper choice of
is valid wit,h sufficient app
smoothing out value e (a, .e . the que,nti tie s T, X, Y Z in the formulae
s restrictedly fluctuating function.
(J,) or (2) } a
'.s ? property far the significant f unctions
We assume tha
in the dyesmics of the atmosphere, i.e., for the velocity
~. ~t.
is u V w for the . specific volurrt,~; and for the pre
corrtponen a
sure p?
Correlation M nts N
and be any two such restrictedly
Let
f unctionsa We signify as ncorrelation monienttt of the Functions
q#7 and. (./) the mean value
J
7)
correlation moments, constructed in pairs or the
These
various elements, are known to be the significant parameters for
.
the ibution of the various value systems of the
statistical dis tx
ion Go Therefore R ( C4 )
observed function a.tha.n the reg
is the quadra tic of the ttspread" of , while the quotient
yR?m~ Raw, ~
represents the correlation factort' of C~4
and
The 6 characteristics introduced by Reynolds represent
of the 3 velocity components: R(u,u) etcp
the correla,tion moments
For the s bol R V the fallowing commutative and
~i
distributive lawn are. valid:
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R(c1)= &cb
(6)
(9)
contain. ~?~n~~
lntxoduct~.on of
' conceptbon?
Now we come, to a major expansion of Reynolds
We ?ch axe re late d to the sta.ta.s~tical
~
~,ntxo duce ope rata. o ns wh:~ '
neigh
bons of two ng
d.
en the simultaneous con
col~eCt~.ons between than this, between, the
is in space, or, more genera1
baring po~.n
ims."
a
' nuversal p
conditions of two different
t there be two f unctions
Le
' the coordinates. We set-,
of ~,~tie and of
I ( - I ) y - 4)
-,J
(a)
In taking the
lation moment 'R 0
and construct the come. undamen~~ variables
an value of this only the value 5 of the f
me P L , can be regarded as
le the ~.ncr~:~nts
t,X,~'az,vaxy whi
constant parameters.
(b)
We now further sets
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The quantity R(() an accordingly be regarded either
is ~
as a function of the eight argunts t, x,y, z;
or
Y2, 2?
In the first case we signify the function with the symbol. ?
and in the second case with ~ A Therefore the following
definitions are valid: (lp)
Y~ = ~J I ' X') ) l 011 X l) y i ' XZ J l
I ~~rJ M~ l r ~ l ) ~-J ~.
J 7'
as a functIon of the arguments t-, xl, Z1; t 2x 2'
We call the quantities k:1 , ttcol~taining moments",
because a 'containing tendency" of the variations from the re-
lative mean finds expression in these t'moments", to be exact,
this s tttendencY" exists with respect to time as well as space
variations.
Calculation Rules for the Ponta ng~ n,.nt o
The following properties are contained in the symbol
to and
c5LJ
is not commutative with respect
Q Instead there is the relationship
pi(J J fl J r.JJ i
T
; , )
On the other hand, the sane property follows from the
law w for R t~ for the newly introduced symbol
distributive
1Jhen the increments
are reduced to zero,
the containing moment goes into the Reynolds correlation
~i
moment
First
12)
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(13)
'urther, the following differential formulas are valid:
- I a_ z ~d s a d> R~, ~
L7-- = z l ds d a-~
where-s, as before, represents any one of the four variables
t,x,y, z, while Gr represents the resulting increment
respectively )
Proof: From the equation of definition (14) and with
help of (a) and (b) we obtain:
cp(1
I)) - -
On the basis of the distributive law we further have:
~r e ~. t .~
Thus we get the differential formulas
c',1 y1
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(c)
(d)
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and in the same mariner
#? ,
uia (1L) in that we express the deriva'
We arrive at Form
tines of with respect to sl and s2 by dexa.vativos o
w rpect to s and m" . This we do on the basis of the rewith respect
latio ns (10) and (b)0
lr~ con '.on with the relationships developed above, we
,~uncts
?'tional formulas which relate certain mean values to
give two add.t.
'
the symbol yl
Now we write formula (13) in abbreviated formM
From this we obtain, using )
_7 c/ )
" we further get with help of the di.ff erential
1 )
~'rora (5
formulas (b) and (1L):
-:-i; - __
::
y r 1 1 L
a`eiA
3 ()+ (A)0
Suementary Assumptions
'.al shown above requires no further hypotheses beyond
The mater
the assumpt'ona that are the basis of Reynolds' theory. Now we must
~.
make two further limita tionS. Without these imitations our algorithm
for productive treatment of the hydrodynamic equations would not be
sufficient. This can be seen in that we cannot operate with expressions
calculation rules we have set up thus far.
of the farm using
First, we must postulate that the containing moment can only
assume values noticeably different from zero in the immediate neigh x
) c7 r- I c
boyhood of point (t, s9 Yj ) let us say in a small value
r7 7"
region 'of the incrc-rents f
k
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ion r all smoothed-out
H*
t must be o! t - the e" 'e
eto. can be handled aco s,. tso (This is
f u11ion
and makes a demand similar to the one made in
Postulate T],
Postulate l for the region 0)
Secondly, .,.we assume that apart from of the
.
t and their derivatives we can overlook moments of
higher order. (This is postulate iii.)
Such 13 the case a$ soon as we admit that the "Smoothing
out values'1, which are forrrled by the process of taking the irean
of snnoothed-~out oscillations C
can be observed
as a s stem of waves not only of short..periods (as is demanded
by postulates l and ii) but also of small amplitude.
On the basis of the last postulate we can set up appro-
ximation formulas for the calculation of mean' values and of
correlation moments for combined expressions which depend on
several variable functions?
Without going into these general formulas further, we
show a very special expression which we need for our immediate
purposes;
proof Let us introduce the correlation moments of the
third degree in which we sets,
R (tn, Q'.1 3)=(:
) ''2-~22' (p23- )
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and thus with the help of the Reynolds' calculation rules we can
easily verify the following exact formula;
&/j)3) )*
If we neglect the third term on the right on the basis of
the above postulate, then e have the approxinte formula (17)w s
Farmul.a (17) gives us the transformation we have been
seeing for the expression ~ on the basis of the definition
a S
the containing moment according to Formula (10) with the help
of
of Postulate TT and the caf f'erential formula (]JJu)
(18)
L / 1 ...
~ a1
These formulas are obviously a generalization of the for
alas (la) which we receive again if we set f In (18)
Setting P the Differential Equations of Turbul.E~nt Motion_
, -
__~,__u__s___....,
The operations mentioned above will now be applied to
thehydradynami.c equations,
Let us at first limit ourselves to the case of the adia-
bath, ..motion. of an Ideal heavy compressible fluid. Let us write
the equations of motion in the Finer forn1t
10.
Declassified
r rc !r n 1 C u
~~ ~.. ~?p;,~III h~,?I d~4N,u41'.rJ~!~7~"a`~I~I~1~PGi~;~?~LYI'~~~n;
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- -
w___
tt
G~GJ _ u1 / (7X 70~'
Q~
(19)
it eastel' to have a perspeCt :VV of the
In order to make
s we will designate the five uriknawrl
following abserva ta.on ,
ns , u,vsw,t, p re sp e c Lively by + ,f ~ , ~p a )
fltnGtla and
will write the five equations (19) in the f oxm
a
or' by setting
t
(20)
we can write even more briefly:
0
(20a)
We will now try to transform these equations so, that in the
(/C ((' !
esulting system th 1functb0n s no loner occur and we
r
have s instead, the an values and the characteristiCS of the
turbulence K9 q4
w.
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tion with this last statement, we want to prove
In ccannec
that there Ccharacter'a sties satisfy the initially se~ conditions
. a
f or a complete system of characteristic.
sforma' 0n mentioned we have the use of
For the Iran
]:e taking of an values, (2 (1) the ~ ~;~-
two operations' . ( m~?
the formation of containing moments.
eneve r a. 't ' s not necessary to make particular indicant
~1h ~.
Lion of the functional dependence
of the quantities
on their eight
r the
arguments, we wil:L replace this symbol with th
thetical expression (i W~
We now form the following equation
simple paren-
equations)
f , )m'C (2 equations)
(21)
(22)
0 25 equations) (23)
( c: PkJ .~
calculation rules mentioned above
using suc:cessa.velY the
can obviously be transformed in the man-
the equatians (21) (23)
ner we wish.
The resulting equations then contain the following quan'-
tide s . (l) the ~ flue mean value o ~~ and.. the de riv~, five s of these
: G
quantities ~ltspgot to time and the coord~_na~.~5o
~-es _ with re
2 the 2 containing rnornents ( P
424 ) and their
with respect:to the time and the coordinates'
derivatives, first
l2 ..
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w ( i ) ) I 2)
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with respect to the increrriantsi1 '
secondly w~,
( 3) the special values, which the functions mentioned in
(2) and their derivatives assume far the special value system
0
The functions shown in (1) and (2) must also satisfy the
1'oi1c$ring supplementary conditions:
(a ) the five quanti tie s ( depend solely on the time and
the coordinate s, but are independent of the incrernen.ts This
dives us x 14 , 20 equati..ons of the form;
(b) (@L:)?) are made subject to the condition (11) which
Produces 2 finite relations among 50 functionsf
apt j ~,~ ~ ~a )
X) I~ d and ~. ? ..E1 ... 2- r) A . These relations
we write in brief as follows
Conversion and Discussion of the Equations Produced
(2)
In order to make the connections between the equations
(21) - (2~) clears` we write the hydrodynamic basic equation
(19) in the form (20) From this the equations (21) W (23) can be
thus represented:
at
13
(22a)
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(23a)
We dc. ave top th,e two parenthetic expressions on the left accord'
lng to the differentiation formulas (ili.) and then, after addition
and subtraation~ we get;
) )= )? (') )
_I__ , ) ( ) ) - (p ) )
(27)
(28)
Since the expressions F, etc. do not contain derivatives
~.
with respect to t9 then - on the basis of the generalized
di.fferentiation formulas (18) - the expressions on the right
in ( 27 ) and ( 28) ca ,nnet contain derivatives with respect to t
41 ith respect to x,s,The system
or ~ but only those w
(27) then represents the result of the elimizaation of all
differential quotients ~si~g"'wth respect to from the
equations (22) ? (23). The system (28) is a result of
the differential quotients with respect to t.
the elimination of
From thi s we may conclude the ini tial value s of the
ontalfl..ns moments (( ) ~produced by time paint t 6 can
not be observed as arbitrary functions of x, y,
c~ 4) most de f ina,.tely
since these values must satisfy the equations (28)
under any condition.
The systems of 'equations (26), (27), (28) is equivalent
to the system (21) (23)Therefore in order to determine the,
~:: 1 `r , ,.,li Eai =f f .P
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we can held an to equat~,ons (26) ~ (28) and
tae
~uncta.ax~s sought
the seconda'Y
o ction wjth the latter it can be show11' that it is
In cazln
aa.tion5 as the Gonda.~~ons to be
.~~ dent to intx'oduce 'these con
1
sv.f ~ a.
~ the unknawnc ons
' o se d on the ini ~ta~,~'ue.....~. s - ?
~n~p
, mo7aent
Let the~3 be given for the a.TU.ti8:l.
e iven the values of ~Pe
an o~~e time point and these
rally spealcing, for any
~, ~ 0, a r ~ ene ~L there be given the
be Functions of x,y,Z' e r'
values shall
f a deFinite value of
( /1 ( \ for the same ' me poa.nt,9 or
~ }' Accord-n~lY
~ a
,I_ \ as fuTictions aF , Y Zi
.~' ~ satisFied. Fux'"
For the initial values ire
the condi.ta.pns (2)
sfY the cOnda.ta.anS (2~)
that, they must ..also -rata.
W tern of the ~~ equations
With these assu~rrptions the s~',~
~
1 system for which the uchY Pr.?oblem
( 28) repre cents rrra
can. be a~~ 1Ca nctioxas ('La ra be
i.e? the 25 mown f U
solved,
eq~~,tion" as ~'unct~-ons of the
.fined on the basis of these
a,,cPrta
7 argW nts X,y,zJ , and
nt we will call Operat~ ~~ A.
This ascer~.~.a-tee
n?tite s (~~ i )a -~G,
d the 30 u~raown q'
han
other on the ua~tions (27)
ue;~through the 25 e~'
From the same i~.tial va7-
coming an be ~,epresented as
the equations (26), c
wa.th the aid of
cta.onsof the gnd t. This we will refer
f~ . ar umex~ts a
to as O uat~-ons (26) (~7)
B (`the system of the 30 eq
. ,.- o --
iffers from a norn~al system of ' 'f,erential equations in the
re ssions the value s
at in the righ,t~hand eXp
ordinary sense in that
eCt to
e ,known functions and their derivatives with rasp
of nth
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the general value system )1 as well as for
the ;r), QC~11r :or th
the special value system : = w a. This condition does not make any difference, however, for the setting up and solution
ai t his problem analogous to the Cauchy Problem.)
'
On the basis of the proceeding observation we can riovi
make the statement: if a value system exists at all for the
desired 30 fuYicta.onS of the 8 arguments t, x, y, z)^Ld) ~ , which
satisfies the equations (2L.) - (28) and which takes on the
initial values demanded for t = 0, t ~0, then this value system
can clearly be found through successive application of Operations
It makes no difference in what order the operations are
A dnd B
applied. (It is not desired to assert that the initial valuef
the 2 containing moments actually cart be -treated as arbitrary
functions independent on one another. The opposite statement
can be proved in the following manner. If a system of these
quantities function of the 8 arguments
quantities exists as a
which satisfy the equations (27) and (28), then the following
2 equations must exist:
If these expressions are developed and all the differ-
ential quotients with respect to t or are replaced by their
expressions from the equations (26) - (28) then we get a system
of equations of the second order which only contains derivatives
~ equation of
with respect to the 6 argumentsx,y,z, f, Each
this type prcduce5, however, one of.the conditions which were
placed on the initial values (valid for t.
h
q j
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(.,
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Only when the conditions so constructed are actually
satisfiedi does the use of the Operations AB and BA produce one
and the a same result for the whop value region of the 8 argu-
ments).
If we are not further interested in the determination of
the characteristics of turbuJ..ence and are only interested to
ascertain the quantities 2, then we can forget Operation A..
From this standpoint we can be content with the determination
of the containing moments for the special value ,G =o (in the containing morne~~t case
r ~ ~. Thus we ge ?t a compl.e to
of the unlimi to d variable
system of characteristics which, with the exception of the co-
ordinates and time , depends only on the three further argumnts
relation (2) the system is reduced to 15 basically different
functions.
1 We should like to remark that in view of the
We must also prove that, as soon as the conditions (24)
and 2 for the initial values t=t=0 are satisfied, the condi-
tions also have validity for the region enlarged by the operations
A, respectively B.
We need only say that for the equation (21i.) the C remain
~'
unchanged by the operation A and that, on the other hand, the
express which are on the right side of the equation (26)
~.onc ~ F~
are independent of the arguments '. From this, it can be seen that
our assertion in connection with operation B is corrects
To prove the resulting assertion for the relation (
it is enough to show as far as these relations together with
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ar e val a,d ? or a ragi on liIIIi. by do tax'
the equata,et ns (2). )
ninatiofl of a definite pair of values 'or this region them
are theuowing equati?ns t'
c~
}
'ons an extension of the relation (2~) takes
through these equati
place beyond the region in questiono
We arrive at these equations in the following manner: First,
the differentiation of the equations (?~) results in:.
.
(DSC~PW)l-- ds <
where the quotients contain, besides the quantities u, c:;3 . .
the tree further quanUtieS R ) d
We have then 9 equations for 12 unknowns and we therefore
do not get a complete system in this ways The number of iride-'
finite functions remaining' has been reduced considerable in can-
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Declassified in Part - Sanitized Copy Approved for Release 2012/04/20 : CIA-RDP82-00039R000200040024-0
I ~Tn~ 1'~ 1" 4.4h
 Declassified in Part - Sanitized Copy Approved for Release 2012/04/20 : CIA-RDP82-00039R000200040024-0
tract to the results of the simple inean~taking that is, the
results o!' Reynolds' approoache
We can achieve a complete norrral system from the sug-
gested system of equations (32) if we arbitrarily' control the
~G (p l~) e In this manner we
3 functions Ik (J Rc
P
have obtained by integrating, as a particular solution the fo1~
lowing case of turbulent motion;
)):: 'P(Xd-
R(f P)~ max? ~,~d%E+aa-b~'~x +
e
rr ~~
/Q)< (p1ti)
z
rewMMA
RCw,p~?
?o i
t)
(33)
(C
C,a,b,c,, a~ are arbitrary constants in this case and
is an arbitrary function of their arguments.
The fluid behaves, in a given case, like an incompressible
fluid of the density 1, in mean motion. The mean motion represents'
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 Declassified in Part - Sanitized Copy Approved for Release 2012/04/20 : CIA-RDP82-00039R000200040024-0
e~ a unifoxm tranelatiQn of the fluid
~ixldeT the se ca.rcumstarlc
as a rigid body
As we have noted, the equatian~ valid for the character~
a.sta.c Rs are not a complete sysbemo The ma.ssing re1atiQnsbap
' s turbu7~erlcemu~~ be determaned
~.c
between the charact,er:~st of the
experimer1tallyp
END--
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