On wave properties of an incompressible turbulent fluid

Physica A 168 (1990) 881-899 North-Holland

ON WAVE PROPERTIES OF AN INCOMPRESSIBLE TURBULENT FLUID O.V. T R O S H K I N Computing Center of the Academy of Sciences of the USSR, Vavilova 40, Moscow, USSR Received 31 January1990

The problem of the propagation of small disturbances of the averaged field of velocities, Reynolds stresses and the pressure in an incompressible turbulent medium, with negligible molecular and turbulent diffusions, is considered. The availability of a limit velocity of such a propagation and the existence of a characteristic cone similar to the Mach cone are discussed. A characteristic equation analogous to the equation of "C±-characteristics" in gas dynamics is derived. The calculation of the geometry of the initial part of a plane turbulent jet running out of the channel into a rest liquid is carried out. Angles of slopes of straight lines of internal and external boundaries of the mixing layer of a jet near the exit section of a channel are related to distributions of steady velocity and Reynolds stresses in the channel. The formal analogy existing between waves of small disturbances of an inviscous and incompressible turbulent medium and electromagnetic waves is established.

I. Introduction A p p a r e n t l y , it was T h o m s o n w h o first o b t a i n e d a w a v e e q u a t i o n for an i n c o m p r e s s i b l e fluid b y a v e r a g i n g E u l e r ' s e q u a t i o n s (see refs. [1], [2, §20]). A t all e v e n t s , a c u r i o u s t h e o r e t i c a l fact c o n s i s t i n g in t h e m y s t e r i o u s h y p e r b o l i c i t y o f a q u a s i l i n e a r d i f f e r e n t i a l o p e r a t o r , with p a r t i a l d e r i v a t i v e s o f t h e first o r d e r , p r e s e n t in a p p l i c a t i o n s o f t u r b u l e n t m o d e l s c o n t i n u e s to a p p e a r in p u b l i c a t i o n s (cf. refs. [ 3 - 5 ] ) , a n d it is t i m e to give s e r i o u s c o n s i d e r a t i o n to it. T h e scalar field o f a p r e s s u r e p d i v i d e d b y a c o n s t a n t d e n s i t y a n d t h e v e c t o r field o f v e l o c i t i e s u = (ui) = ( u l , u 2, U3) , a s f u n c t i o n s o f t h e t i m e t a n d o f t h e s p a c e v a r i a b l e x = ( x i ) , a r e b a s i c p h y s i c a l fields d e s c r i b i n g t h e m o t i o n o f a viscous i n c o m p r e s s i b l e fluid. T h e c e n t r a l d i m e n s i o n l e s s p a r a m e t e r o f such a m o t i o n is t h e R e y n o l d s n u m b e r R e > 0 d e t e r m i n e d b y t h e r a t i o o f t h e p r o d u c t o f c h a r a c t e r v a l u e s o f t h e size a n d t h e v e l o c i t y o f a s t r e a m to t h e coefficient v o f a k i n e m a t i c (or m o l e c u l a r ) viscosity. T h e m o t i o n is fully d e s c r i b e d b y i n t r o d u c e d fields, for c o m p a r a t i v e l y s m a l l v a l u e s o f t h e n u m b e r R e . F o r sufficiently l a r g e v a l u e s o f R e , t h e b e h a v i o u r o f fields p , u is r a n d o m . T h e last is f o r c e d to h a v e r e c o u r s e to t h e a v e r a g i n g of v a l u e s o b s e r v e d a n d to t h e 0378-4371/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

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o.v. Troshkin / Wave properties of a turbulent fluid

extension of a set of values averaged. The average f of a value f depends linearly on f and conserves a non-random value (particularly, f = f ) . In the widely used approach introduced by Reynolds, the extension of the set of averaged fields p, u means the addition of one-point correlations ~-i~-= ui(t, x) u~(t, x) = ~i, i, s = 1, 2, 3, of the pulsating component u' = u - ~ of the velocity u to them. The correlations % generate tensions called "Reynolds stresses". The set of them form a symmetric matrix field 3 - : (~'i~) called the "Reynolds tensor". The medium is considered to be a "laminar" one if 3-=-0 or to be a "turbulent" one if 3-7-:~0. In the last case, the quadratic form of the matrix ,~Yis supposed to be a strongly positive one (that is responsible for the essential three-dimensionality of turbulent pulsations). The governing equations of a turbulent medium are the same Navier-Stokes system, however, considered on functions that take random values. The averaging of these nonlinear equations yields new additions of the form V ' T i : (O/OXk)Tki = (O/OXi)Tli ~- (O/OX2)T2i ~- (O/OX3)T3i. One can then derive the evolution equations for Ti~, (see section 2 below). But the system of Reynolds equations for coupled fields ~/, 3-, fi will contain new moments hi,` of the averaging and so will unclose. Plenty of theoretical works are devoted to the problem of the closing of the Reynolds system. In many of them, "unclosed" additions hi~ are approximated by semi-empirical relationships modelling the physical processes of turbulent diffusion and relaxation of correlations ~'i~ [6]. Besides "unclosed" terms hi~,, the Reynolds system contains "exact" quasilinear additions with partial derivatives of the first order depending only on functions di, %,, l~ and present in all closure schemes of the type mentioned above. From the formal point of view, the work done is devoted to the study of the structure of a matrix differential operator generated by these additions. A scheme of such a study will be similar to that used in gas dynamics. By disregarding of terms of molecular and turbulent diffusions, we find characteristics of the corresponding " t r u n c a t e d " Reynolds system and analyse linearised equations. Particularly, we show that the availability of turbulent tensions in an incompressible medium leads to the arising of oscillations propagating with a finite velocity in the form of elastic crosswise waves analogous to electromagnetic waves. On the whole, the present work is a developed variant of refs. [11-13].

2. Truncated Reynolds system Let us outline the following model of an ideal (i.e. inviscous and incompressible) turbulent fluid. We shall consider a continuous incompressible medium

o.v. Troshkin / Wave properties of a turbulent fluid

883

submitted locally to Newton's dynamical law, or satisfying Euler's hydrodynamical equations: D,ug + u.Vug + D i p = 0 ,

(1)

V- u = 0.

H e r e D, ~- O/Ot, D i =- O/Oxg, u "17=- UkDk, 17" U =- D~ug ==-div u. By repeating indices we mean summation. External forces are absent. Further, we suppose that the liquid is turbulised, or that the fields u, p take random values. Substituting u = ti + u', p = ,6 + p ' in (1) and averaging (1), one finds D,ti s + ff .Vffg + u' "Vu I + Difi = 0 ,

17. ti = 0

(l'a)

(since -uv = 0, p-V= 0). Subtracting ( l ' a ) from (1), one has D , u I + 1i .17u I + u' .17~g + u' .17u I - u' .17u~ + D i p ' = O,

(l'b)

17" u' = O.

(l'c)

Multiplying ( l ' b ) by u;, exchanging i and s, summing the two equalities obtained, averaging the sum and accounting for ( l ' c ) , one arrives at t

¢

--

!

t

i

!

!

t

D t u i u s + U "17UiU s -~ blil, I "17ffs q- tlsU "17~i q- hgs = 0 ,

(l'd)

where hi s = u ~ D s p ' + u ; D i p ' + 17. ( u ' i u ; u ' )

(l'e)

(we use the identities u'iu'.17u',+ u ; u " V u i = u " V u ' i u ; = 1 7 " ( u l u £ u ' ) ) . Eqs. ( l ' a ) and ( l ' d ) form the Reynolds system, with v = 0 . (To obtain the full Reynolds system, with u ¢ 0 , it is necessary to add the "viscous" term v Aug = - uDkDkU i to the left-hand side of the first equality in (1) and repeat the procedure conducted. The left-hand sides of eqs. ( l ' a ) and ( l ' d ) will then contain additions - v At/g and - v Au'gu's + 2v17u~ . 7 u ; , respectively.) Finally, following the closure schemes for hgs [6], we approximate hgs by the sum -

hi s = _V.

(k17Tis ) ~_ ol(,l.is -- T is-(°)'t) ,

(ltf)

where k and a are coefficients of the turbulent viscosity and of the relaxation, respectively; if(0) = ,~q'is (0),) is a prescribed correlation matrix. We neglect the

884

O.V. Troshkin

/ Wave properties of a turbulent fluid

turbulent diffusion: (l'g)

k=O,

and suppose that a > 0 is a constant. Using the identities u' .Vu I : 7 - ( u l u ' ) : 7-~-i (following from (l'c)), one can rewrite the results ( l ' a ) , ( l ' d ) , ( l ' f ) , ( l ' g ) , for an ideal turbulent fluid, in terms of vector-columns "r. : L/~/J'

of the matrix 3 = (z~), in the form of the following "truncated" Reynolds system: D tt~ i + 1~ " ~Ui

-~- V *

"t"i -~ D i]5 : 0 ,

D,%, + d-V%, + %.Vr7 + % .Vr~ + h~, = 0 , V. d = 0 , h,.,

(2)

= O~(Ti, , -- "i'l;))) .

3. Characteristics of the truncated system

By the characteristics of the system of quasilinear equations with partial derivatives of the first order considered in the space of variables t, x one usually understands a smooth surface of a level:

q (t, x) : const.,

(D,qO" + Iv l 2 0,

Iv l

such that the system under consideration cannot be resolved uniquely concerning the vector of derivatives of unknowns taken in the direction of the normal W to the surface ~(t, x) = const, at any point (t, x) of that surface. It is obvious that the derivative with respect to W is proportional to that taken with respect to the variable ~. So the condition is equivalent to the following restriction on q~(t, x): there is a nontrivial solution of the corresponding linear homogeneous system considered for derivatives on the variable q~ and obtained by the linearisation of the initial system and by the subsequent removing of derivatives on local coordinates supplementing the variable ~ and of free terms (containing no derivatives). This restrction means that the determinant of the linear homogeneous system considered equals zero. This determinant is a polynomial of Dtq~ and of components of the gradient 7q~. So the condition for a characteristic ~(t, x ) = const, is reduced to a finite number of elementary

O.V. Troshkin / Wave properties o f a turbulent fluid

885

algebraical equations for partial derivatives of ~, or equations determined by roots of the polynomial mentioned, so-called characteristic equations of the initial quasilinear system. We prove in section 7 that the system (2) has the following characteristic equations: Dt~ + t~.Vq~ = 0,

(3)

IV~I~0,

and

Iv ol = 0 ,

D, p # 0,

(4)

exhausting the set of characteristic equations admitted by Euler's equations (1) (the case of eqs. (2) in which ff-= 0 and a = u, fi = p) and, in addition, for a strongly positive definite symmetric matrix field J-, the equation (D,~ + ff.V~o)2 = (~-V~, V~),

Oq~ Oq~ (3-V~, V~) --- ri, ~x~ ~x, '

(5) Dt¢ + a'V v # 0 ,

IV I#0,

completing, with (3) and (4), the full set of characteristic equations of (2).

4. Wave characteristics and beams

Eq. (5) determines the phase ~ of some wave of small disturbances in an ideal turbulent fluid, or of a "turbulent wave". The corresponding "wave characteristic" ~(t, x) = const, forms the front of a turbulent wave. As follows immediately from (5), for a prescribed vector y, lyl = 1, the front of a plane turbulent wave moves through the medium with some relative velocity 0 < Cy 0 , and the "main velocity" is cy = X/A. For the turbulent jet injecting into a stream, in the direction of the stream y = (1, 0, 0), c r = ~ 1 equals 0.65, 0.6 and 0.45 m/s at a distance of 50, 100 and 150 diameters from the source, respectively; the velocity of a jet is 10.2 m/s, while the velocity of a stream is 8.5 m/s [7]. Introducing the average velocity

( Url2-[- U ,22 q- ul c=

3

1/2

(6)

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O.V. Troshkin / Wave properties of a turbulent fluid

of the relative motion of a turbulent wave, one has, for the atmosphere, that the ratio c/a, where a is the velocity of a sound wave, is approximately equal to 1/300 [81 . In the stationary case, for a homogeneous and isotropic turbulence, when % = c26i.,. (where 6i., = 1 for i = s and 6~,.= 0 for i ¢ s), eq. (5) takes the form (ff.V~o)2=c2lVq>] 2, or, in the coordinate system where the vector f f ¢ 0 is collinear to the axis x,, lal2(a,~/axl) 2= c21V l =. Introducing the number M,.= I l/c (analogous to the Mach number M , = lUf/a), one nds that the equation obtained has only the trivial solution ~ =-const. if M, < 1. For M C= const. > 1, the equation admits the "characteristic" cones ¢ =x~ ~" V ~ - ~ - 1 I ~ f ~ - x~. So eq. (5) reveals a p h e n o m e n o n similar to the "transsonic" effect. The dagger blade form of the turbulent kernel arising from the broom-like form in the initial part of a jet is explained, apparently, by the p h e n o m e n o n shown in fig. 1. Using eq. (5), we shall find the slope angles /3+ > 0 and /3 < 0 of external and, respectively, internal boundary straight lines of the mixing layer of the initial part of a plane steady turbulent jet running out of the channel into a rest liquid (see fig. 2). For this purpose, we shall relate the angles/3+ to quantities characterising the steady turbulent flow in the channel. The boundary lines mentioned are determined usually [9] by those on which the approximate profile of the prolongate velocity component U has a discontinuity of the derivative with respect to x 2 (see fig. 2). We identify these lines of

II,

Me < 1

F/c > Fig. 1. Turbulent "transsonic" effect.

i

0

Fig. 2. Prolongate velocity component near the exit section.

O.V. Troshkin / Wave properties of a turbulent fluid

887

a "weak discontinuity" with wave beams determined by eq. (5) considered for q~ = q~(xI , x2). Each such beam passing through the point (X°l, x °) is a straight line: o x2 - x2 - k 0 ~ X 1 -- X1

k --

Oq~/OXl

o o ( X l , X2)

O~/~X2

0 0 touching t h e c h a r a c t e r i s t i c ~0(Xl, x2) = c o n s t , at t h e point X 1 -----XI, X2 = X 2 a n d approximating it near this point. For 5 = (U, 0, 0) and ~0 = q~(Xl, x2), eq. (5) takes the form

OXl aX2

\axl/

22\~X2 ]

= 0

and has solutions

k = C_+ =

--T12 -I- VT~2 --~ ( U 2 _ Tll)T22 U2 _ TII

So, for slopes C_+ = tg/3+ of beams, one has (cf. fig. 2) T12

It/(T12~ 2

-- U ~ -¢- ~ \

(7_+ =

U2 ]

( Tl1~ Tll T22 -~- 1 - U 2 ] U2 TI1

rli U2

(7)

Formula (7) gives the relationship we are interested in. Further, for a steady turbulent flow in the channel, the functions U, rlt, r12, '7"22depend only on x 2 [10]. So C+- = C+-(x2). The beam forming the external boundary of the mixing layer passes through the point on the wall x 2 -- 1 (the variables considered are dimensionless). The beam of the internal boundary passes through the point x 2 = x~ that determines the position of the external boundary of the boundary layer on the wall (see fig. 2). The point x~ is determined approximately by positions of the maximums of the functions %1, %2, %2, for 0 < x 2 < 1, measured in the experiment described in ref. [10]. Both points are placed on the exit cross-section of the channel. According to ref. [10], for x 2 = x2, * x/--~I/U=~/12, x/'~22/U=l/12, rlz/X/-~H x/-~22=0.4; for x2--> 1, %2 = °(U2) and x/'~11/U ~ 0.3. The work [10] contains no data concerning the quantity y = lim v'~22 x2---*l ~ "

888

O . V . T r o s h k i n / W a v e properties o f a turbulent f l u i d

A s follows f r o m ref. [10], 7.22 < 7"11" Supposing the existence of the limit value y, o n e then finds that 0 ~< 3' ~< 1. Substituting the data o b t a i n e d in the f o r m u l a for C+, one has 0.3y C+(1) = lim C ~ ( x 2 ) = + - 0.4 2V2~.5 ~ / 0 . 4 2 " 2 . 5 (2.5) ~ 12 --+ ~ 12 4 + 1- ~ G(x

) =

So - + C ÷ ( 1 ) > 0 ,

1 i2 2

2.5 12 2 -+C_+(x~)>0, and we find

0.3y /3+ = a r c t g C + ( 1 ) = arctg ~

,

/3_ = arctg C (x~) = - 5 ° .

F o r the largest admissible value 3' = 1, one finds /3+ = 17 °. T h e e x p e r i m e n t described in ref. [19] gives us/3+ = 12°-16 °, /3_ = - 7 °.

5. Structure of turbulent waves

T h e electrical and the m a g n e t i c c o m p o n e n t s E and H of a plane electrom a g n e t i c wave oscillate in a plane o r t h o g o n a l to the direction E × H of the p r o p a g a t i o n of a wave. As we shall show, t u r b u l e n t waves have an analogous " c r o s s w i s e " structure. We shall consider an ideal t u r b u l e n t fluid in a dynamical state of if, 3--, 16 which is close to constant fields if(o), if(o)----tT.isZ (O)x), /6(0) satisfying the system _(0) (2). Substituting if/ .-(0) + ~i, 7.is = 7.~') + rG, ,6 =/6(o) + ~., with ff~0), Vis lff(o)= const., in (2), disregarding t e r m s quadratically d e p e n d i n g on the functions ~i, ~7~,, if, including their derivatives, and after transition to the m o v i n g c o o r d i n a t e system: t-*t,

_-(o) Xi"'~ Xi -- fU i

(or D/---~ D i

~

D, + if(0)'V---~ D , )

,

one c o m e s , in the case of zgs = c26/,, to the following linear system for small disturbances sc = (~:i), ~7 = (7/i,) = (rls/), ~': (8a)

Dt~i + V. r/i + D/~" = 0 , Dtr/,s + c 2( D /sc, + DssC,) +

his

= 0,

(8b)

O.V. Troshkin / Wave properties o f a turbulent fluid

v.

= o,

889

(8c)

(8d)

his = Ol1~is ,

where c, ~ = const. > 0. Given a constant r > 0, we introduce vector fields E = (E s), A , H, j = (.is) and scalars ~0, p: E s = KV-r/s,

A = rc~,

H = KcV x ~ ,

K

Js = ~

(9)

K

V. h s ,

q~ = K~,

p = ~

D~D k r/ik .

Multiplying then (8a) by K, one obtains the relation existing between the electrical components E, the scalar potential ~0 and the vector potential A of the electromagnetic field: 10A - - - + E + grad ~p = 0 c at

(8'a)

where c corresponds to the velocity of light in vacuum. The identity (8'a) implies the Faraday's law of electromagnetic induction: / 1 3A \ 1 OH ---+rotE=rot!---+E+grad~0!=0. / c Ot

\c

Ot

Acting on (8b) with the operator (K/c) div (or differentiating with respect to xi, summing over i and multiplying by r / c ) and taking (8c) into account and the identity D i D i ~ - - A ~ = - r o t ( r o t ~) (follows from (8c)), one comes to the Maxwell law relating the conduction current j to the displacement current: 1 0E c Ot

rotH+

4"rr c j=0.

As a consequence, one has the following Maxwell equations for the electromagnetic field in the vacuum: 1 OH ---+rotE=O, c Ot

10E cot

divH=O, (10)

rotH+

4"rr cj

O,

div E = 4arp.

Following from (8d), Ohm's law j = irE,

tr = 4"rr

(11)

890

o . v . Troshkin / Wave properties o f a turbulent fluid

(obtained by acting on both parts of (8d) with the operator (K/4~r) div) completes the system (10). So, the propagation of small disturbances of fields if, 3-, fi in an inviscous and incompressible fluid really obeys laws of the electromagnetic field.

6. Thomson's hypothesis On the other hand, the considerations described can be used for the concretisation of Thomson's hypothesis concerning the turbulent model of the " e t h e r " , or of a substance supposed at one time to be responsible for the propagation of electromagnetic waves [1], [2, §20]. A modern concept of the ether consists in the "vacuum hypothesis" and deals with the quantum theory of a field. Nevertheless, it would be useful to present some alternative "neoclassical" point of view based on Thomson's hypothesis. We identify a hypothetical substance of the " e t h e r " with an ideal turbulent medium submitted locally to Newton's dynamical law and rested "in an average sense": K(0)=0. The medium has a homogeneous, isotropic and constant Reynolds tensor 3-(o) = ,~ T i s(o),) , T i s(o) =cSi~ 2 , c=const.>0, andaconstant pressure fi(0)= const. Turbulent diffusion is supposed to be negligible:

V.(ulu;u') = o .

(12)

The relaxation has a simple form: u~Dsp,

_~o)~I + u ~, D i p , = a(.ri, _ TiS

,

Ol = const. > 0.

(13)

The hydrodynamical regime u(°) 0 i=

,

rls0 ). = c 26i, ,

c = const. > 0 ,

fi(°) = const.

(14)

is identified with some equilibrium dynamical state of the medium. An arbitrary non-equilibrium dynamical state t~, 3-, fi is supposed to be close to the equilibrium state. An identification of the ether with an incompressible continuous medium submitted locally to Newton's dynamical law means the validity of Euler's equations (1). The fact that the fluid is turbulised means that eqs. (1) are considered on fields u, p subjected to random fluctuations. Averaging of (1) leads to the Reynolds system ( l ' a ) , ( l ' d ) , ( l ' e ) . The use of assumptions (12), (13) leads to the system (2). Linearisation of (2) in the regime (14) leads to the system (8) considered for small disturbances ~:i : ~i - - I=(o) (o) ' Ai ' ~i.s : T i s - - T i~

891

O.V. Troshkin / Wave properties of a turbulent fluid

Concept of an incompressible continuous medium I

Newton's ~ _ ~ dynamicallaw

Euler's equations (1)

!

! I I

Maxwell's equations (10)

tSmall

disturbances of the simpleturbulent state (14)

1

I Turbulent I [ closurescheme ] ~ I ~ .(13) I

Truncated Reynoldssystem

Ohm's law I (11)

(2)

t fieldsp, u

Neglect of the turbulent diffusion (12)

Fig. 3. Derivingof MaxweU'sequationsfrom Newton'sdynamicallaw.

= f - ~ 5 (°) of the equilibrium state (14). The introduction of the electromagnetic field by formulae (9) leads to Maxwell's equations (10). The closure scheme (13) implies Ohm's law (11). The considerations described are illustrated by the diagram in fig. 3. So the field of small disturbances of an ideal turbulent fluid can be identified formally with the electromagnetic field. The constant K in (9) then has the dimension of m/q where m and q are some mass and the charge, respectively. In the case of a homogeneous and isotropic turbulence, the quantity c in (10) can be determined by the formula (6) considered for u'~ = 7_(o) ii . As follows from (6), c depends only on pulsating components u~. For a given vector field a = 5, we then have that (u + a)' = (u + a) - (u--+-a) --- u - 5 = u', and c is an invariant (it does not depend on the moving coordinate system). So the turbulent approach suggested is consistent with Einstein's postulate concerning the independence of the velocity of light of the choice of an inertial coordinate system. Finally, in addition to (10) and (11), we shall derive the law of the motion of the elementary charge q of the mass m in an electromagnetic field E, H generated by the "ether". For this purpose, we shall suppose that any particle is transparent for the ether, so mff is the potential energy of a test particle of the mass rn immersed in the fluid. We suppose also that each particle floats in this fluid, so, if u is the velocity of the motion of the particle of mass m in the ether, then ½ml~ + o[ 2 is the kinetic energy of the particle (see fig. 4).

O.V. Troshkin / Wave properties o f a turbulent fluid

892

Fig. 4. Motion of the elementary charge in the "'ether".

Supposing that I~l "~ Ivl ~ c (the average velocity of the disturbing fluid is far less than the velocity of a particle, but the average velocity of turbulent pulsations is far m o r e than Ivl) and neglecting the quadratical term l m I ~:l 2, one has the Lagrangian of a particle: L ( t , x, v) = ½mlv[ 2 + m sc" v - m ( ,

where sc = so(t, x) and r = ~'(t, x) are considered to be prescribed functions. T h e m o t i o n of a particle is described by the equations d(OL\

OL

d~ ~,~-o J = ~ x

d

'

0

d t -= ~d + v . V

,

or, since OL/Ov = m y + mE and OL/Ox = m V ( ~ . v)]o= . . . . t. - mV~, dv O~ m - ~ + m ~ + m ( v - v ) ~ = mV(~-~)1 . . . . . . , - mV~.

According to (8a), 0• = - v . ~ at

- v~

(v.~

~ (v- ~i)).

Since v x ( v x ~) = v ( ~ : . v)l~- . . . . , - (v . v ) ~ : ,

one comes to dv

m - ~ = m V . ~? + rnv x rot ~:

(rot ~: ~ V x so).

We then identify mV-~? with the electrostatic force q E , or set the constant K

O.V. Troshkin / Wave properties o f a turbulent fluid

893

in (9) equal to r e ~ q : K

~

m q

- -

.

Using (9), one obtains the required equation of Lorentz: do

q

m -~

(15)

= qE + - v X H

So the basic equations of classical electrodynamics (10), (11) and (15) can be derived formally from Newton's universal dynamical law as acting in an ideal turbulent medium, and Lorentz's force ( q / c ) o x H is generated by the Coriolis force m v x rot ~. 7. P r o o f of the s t a t e m e n t on characteristics

for ~ , ri,, fi, respectively, in (2) and disregarding terms quadratically depending on small disturbances ~i, rl~, = ~Ts~ and ~', including their derivatives, one obtains the linearisation of the system (2): Ui + ~i, q'is + ~is, fi + ~

Substituting

D,~ i + ~ .V~i + ~ .VKi + V. rli + Di~" = 0 , Dtrli , + t~-Vrl,, + ~ .Vr~, + r; .V~, + ~i .Va, + r, .V~ + rb .VK~ + hi, = 0 , (2')

his : otrli s •

For Ui ~- 0 and ris = C28is, C = const., one arrives at (8). Now, for completeness, we shall consider the case of arbitrary dimension n/> 2: i, s = 1 . . . . , n. Following section 3 and substituting the relationships ~i(¢), %(q0, ff(q~), q~ = q~(t, x), for ~,, ~s, ~, in (2'), one obtains, removing free members, the required homogeneous system for derivatives ~i---d~Jd~0, ~is --- d%/d~o = 41s~, ~ ~ d~'/dq~. In abbreviated terms of q~,~- D,~o, q~i--- DN~, q~a --- D,q~ + t~.Vq~, it takes the form + +

V¢. ~ = 0.

+ .v

(2"a) )L +

= o,

(2"b) (2"c)

894

O.V. Troshkin / Wave properties o f a turbulent fluid

The condition for which the system (2"a-c) has a nontrivial solution ~i, 41i~, ~ at a fixed point (t, x) will determine a characteristic equation of (2') (or of (2)) at this point, while the solution ~i, /h,, ~ will determine a respective characteristic direction. The largest number of linearly independent characteristic directions of a characteristic equation will determine the multiplicity of the equation. The analysis of the system (2"a-c) will consist of three parts. In the first and the second parts, we consider the case in which the matrix 3-is nondegenerate. In the first part, we show that each of the equations (3)-(5) is a characteristic one, i.e. the validity of (3), (4) or (5) provides the existence of a nontrivial solution of (2"a-c). In addition, we estimate multiplicities of characteristic equations. In the second part, we prove that (2') (or (2)) has no characteristic equations distinguished from (3)-(5), i.e. the existence of a nontrivial solution of (2"a-c) implies the validity of (3), (4) or (5). In the third part, we consider the case of ,Y-=-0 when the initial system (2) coincides with Euler's equations (1). As for the first part of the proof, one notes that the condition C~ + [V~¢I2 # 0 imposed on the characteristic surface c(t, x ) = const, is equivalent to c

+lvcl2

0

( c , - - - c, + a . v c ) .

(16)

Really, C~ + IVCl2 = 0, or ¢, = q~, . . . . . q~,, = 0, implies ¢,, = ¢, + diq~i = 0. Conversely, ~ , + IVCl2 = 0 implies C,~ = 0 and C~ . . . . . q~,,--0. Then C, = C . - ff i~vi = 0 . Let us show that q~,, = 0 (or (3)) is a characteristic equation of (2'). Really, C, = 0 and (16) imply IVcl ~ 0 . Rolling (2"b) with C, (i.e., multiplying (2"b) by ~ and summing over i from 1 to n) and taking into account (2"c), one finds (fiX'C, VC)~ = 0 (where (3-Vq~, V~) -= rikWiCk ). Since Iv l # 0 and the matrix J- is nondegenerate (i.e. ( J V ¢ , V¢) > 0 for Iv l ~ 0 ) , one has ~ = 0. The latter provides the validity of (2"b) and (2"c). (2"a) is reduced to the following system for //i., and ~: v

Rolling the last with q~i, one arrives at (f/Vq~, V~) +

2 = 0, or

= _ Iv

l:

Substitututing the orthogonal decomposition /li = a~ + A(~i / ]Vq~[z)Vq~, a i- Vq~ = 0, or iV~l 2 ,

aikq), = 0 ,

ai, ,

h = const.,

O.V. Troshkin / Wave properties of a turbulent fluid

in the two equalities obtained, one has 1 + . - - + equations for a~k: aikq~k = 0 ,

895

( n - 1 ) + n easily solved

aik = aki

and to the equality ~ = - A . Consequently, the ~oa = 0 is a characteristic equation of (2'). Since the full number of constants aik, A is n 2 + 1, the multiplicity of ~0~ = 0 is n2 +l-(l+...+n)-

(n -

1)n + 1

-

2 Let us show that Iv l = 0 (or (4)) is a characteristic equation of (2'). Really, [Vq~[= 0 and (16) imply q~a ~ 0. Eqs. (2"a) and (2"b) then imply ~ = 0 and 4/= 0. The system (2") is fulfilled identically for arbitrary ~ ~ 0. Consequently, [Vq~I = 0 is a one-multiple characteristic equation of (2'). Let us show that, for C a ~ 0 and Iv~ol¢0, q~2 =(~-V~o,V~o) (or (5)) is a characteristic equation of (2'). Rolling (2"b) with ¢,G and using (2"c), one finds that (~7q~, V¢)q~ = 0. Since Ca ~ 0, one has (~V~,7~) = 0 .

(17)

Rolling (2"a) with ~Pi and using (2"c), one finds (4/V~, Vq~) + [Vq012~= 0. Since Iv l 0, one has, on account of (17), that ~ = 0. (2"a) is reduced to ~,_

V~ "/7, , ~a

~ =0.

(18)

Substituting ~i from (18) in (2"b) and using q2 = (J-V~p, Vq~), one comes to the following system for /G:

(~-,. v~)(v,p- ,is) + (~-~- v,p)(v~ • ,i3 = (a-v,p, v~),~,s.

(19)

As a consequence of (19), (2"c) is valid. Really, rolling (19) with q~iG and using (J-Vq~, Vq~)> 0, one comes to (17). Rolling the first equality in (18) with q~i and using (17), one comes to (2"c). So the ~i, ~ from (18) and t h e / G determined by a nontrivial solution of (19) are forming some nontrivial solution of (2"a-c). We set out to resolving the system (19) at an arbitrary fixed point t = {0), x = x ~°~ where IVq~l~ 0 . Without loss of generality, one shall suppose that x ~°)= 0 and ( a ~ p / O x 1 ) ( t (°), 0)-760. We introduce a new system of coordinates

896

O.V.

X l ' ~ • . . ~ X n'

Troshkin

/ Wave properties of a turbulent fluid

such that

,

_ (o)..

X l = ~(10)X! "~ ~9~0)X2 q- " ' "

q- qJn

__ Xn , X 2t = X 2 , • . . , X ni --

An ,

q~lo) _= aq~ (t(o), 0 ) , Ox~ or (o)

1

q~z

q~(n°)

t (~O(10) x 2

X I - - ~ (10) X ;

~(o)

t

x,~,

¢

X 2 = X 2 , • . . , X n ~---X n •

F o r a n y p o i n t t, x, o n e has

o(x~ . . . . .

x;)

O(x~ . . . .

0

1

0

0

0

= ~(o) ~ 0

(20)

, x,) ...

1

and O~

OX k

0(t9

Oxl

Ox[ Oxk

0 ~o

(o) q; ,,

~)1

O~

_ (o) ,

0x'2

q;1

q~ ~o) ~p(o) g~l +

q~2, • • • ,

I

_

F o r t = t (°) a n d x = 0 ( w h e n q~i = q~l°)), o n e gets O~ Ox'i ({o), O) = 1 ,

Oq~ (t(o), 0) . . . . . 0x~

0q~ (t(o), 0) = 0 0x,',

(21)

F u r t h e r , using the s y m m e t r y zk,~ : zsk, we can rewrite (19) as follows: ~k

O~

O~

O~

OX k O X m ~ m S

O~

0~

+ T~k OX k O X m ~ m i

= ~m

O~

OX k OX m

or

OXp'

0~

0~9

Tik OX k OXp OXq

OXq'

0q2

0~9

0 Xqt

OX m ~ms -~ T~k OX k OXp

OXq

OX m ~rrti :

.

OXp

,

0~

0([9

Tkm OX k OX m

(22)

Setting

ax' OXq Tis OX i

OX s -

z' Pq

and

~is

OXp

! OXq

OXi

OX s

.t ~ ~pq ,

(23)

O.V. Troshkin / Wave properties o f a turbulent fluid

897

one has 0

(22')

Condition (20) provides the equivalence of (22') with (22). For t = t (°~ and x = 0, on accounting (21), one has 0q~ 0~o (t(O),o) , 0~ 0q~ ,o Tkm OX k OXm -~"fPq OXp OX'q (t(°)'0)='r'n(t(°)'0)--='r11>0' and the system (22') takes a simple form: ,o.,

,o.,

TalT~I/3 + T f l " Q l a

:

,o.,

~ll~afl

,

,o =__z' (t(o),0). a0

(22")

Ta~0

The system (22") is easily resolved for ~ ' a . The full set of nontrivial solutions of (22") depending on n - 1 arbitrary constants % , . . . , %, 7~ + "'" + 3/2 # 0 , is •

•, T~l/3~--y/3,

,

TIll=0,

T~pq = "Oqp ,

,0

,0

•, Tal T/31 T~a/3 - ,o7~+---76y., Tll '/'11

2~a<~/3<~n,

p, q = l, . . . , n .

Using (23), one then finds quantities ilis-

OX i OX s Ox" Ox'~ il"t~

(24)

that are forming a nontrivial solution of (19). Really, if ~is = 0, for all i, s = 1 . . . . , n, then, as follows from (20) and (24), one has 9"~ = 0, for all a,/3 = 1 , . . . , n, which contradicts the non-triviality of the obtained solution of (22"). So q~ = (3-V~p, Vq~) (or (5)) is a characteristic equation of (2'), for q~a # 0 and IV~I ~ 0 , and n - 1 is the multiplicity of ~p~ = (3-V~, V~p). Getting to the second part of the proof, we suppose that (2"a-c) has a nontrivial solution ~/, ~is, ~"at a certain point (t, x), and that (16) is valid. Then

898

O.V, Troshkin / Wave properties o f a turbulent fluid

we have only three possibilities for (t, x): (I) G = 0 , Jv l 0; (II) q~,,# 0 , IVq~[ = 0; (lII) G # 0 , IV,el # 0 . Cases (I) and (II) are reduced to (3) and (4), respectively. In the case (III), we have that the vector ~-= ( ~ ) # 0. Really, ~ = 0 and (2"b) imply that the matrix 4/~-(//~s)=0 (since ~,~ :~0). Then (2"a)implies ~ = 0 (since [Vq~I # 0), and the solution ~, 9i.,., ¢ is a trivial one. So ~ ~ 0 . Rolling (2"b) with ~ and on account of (2"b), one has ('//Vq~, V~p)G ` = 0, or (17) (since G ~ 0). Rolling (2"a) with q~ and taking into account (2"c) and (17), one finds IVq~12~= 0, or ~ = 0 (since ]V~] ~ 0 ) , and we come to (18). Rolling (2"b) with q~ and taking into account (2"c), one has q~,(7~ - ~h) + (8-Vq~, Vq~)~, = 0. Taking into account (18), one obtains [ ~ , - ( J - V ~ , Vq~)]~ = 0, or q~ - (~-Vq~, Vq~) = 0 (since ~: ~ 0). So there is no characteristic equation of (2') (or (2)) distinct from (3)-(5). As for the third part of the proof, we consider the case in which g7 = r / = ~) = 0. In this case, the system (2") takes the form

(25) For (3) or (4), the system (25) has nontrivial solutions. Really, for (3), the ¢ = 0 and any vector ~ ~ 0 orthogonal to V~ at (t, x) are forming a nontrivial solution of (25). For (4), ~ = 0 and any ( ~ 0 satisfy (25). For ¢~, # 0 and [V~I ~ 0 , the system (25) has no nontriviai solutions. Really, rolling the first equation of (25) with ~ci and using the second one, one finds ~p,~l~] 2 = 0, or ~ = 0 (since G, ¢ 0). Since [V~I ~ 0, the first equation of (25) implies ~ = 0. So (2') (or (2)) has no characteristic equations in the case of 3 - = 0 , except for (3) and (4). The proof is finished.

8. Concluding remarks

As follows from theoretical considerations conducted the availability of turbulent tensions in an incompressible medium really leads to the arising of oscillations propagating with a finite velocity in the form of elastic crosswise waves of small disturbances. The value of this velocity is determined by components of the Reynolds tensor and of the averaged velocity. The wave has the structure of an electromagnetic wave. Only "generation" terms

are responsible for these waves.

o . v . Troshkin / Wave properties o f a turbulent fluid

899

T h e r e has b e e n i n v e s t i g a t i o n into s o m e i n t e r e s t i n g w a v e s o f s m a l l disturb a n c e s arising in t h e t u r b u l e n t m e d i u m n e a r t h e wall in t h e e x p e r i m e n t [14, 15]. C o r r e s p o n d i n g t h e o r e t i c a l r e a s o n s [16] a p p e a l to t h e t u r b u l e n t viscosity a n d differ f r o m t h a t p r e s e n t h e r e .

References [1] [2] [3] [4] [5] [6]

W. Thomson, Phil. Mag. 24 (1887) 342. H.A. Lorentz, Aether Theories and Aether Models, H. Bremekamp, ed. (Leiden, 1902). P. Bradshaw, D.H. Ferris and N.P. Atwell, J. Fluid Mech. 28 (1967) 593. A.F. Kurbatskyi and A.T. Onufriev, J. Appl. Mech. Tech. Physics 6 (1979) 99 [in Russian]. D.N. Zubarev, V.G. Morozov and O.V. Troshkin, Dokl. Akad. Nauk SSSR 290 (1986) 313. W. Frost and T.H. Trevor, eds., Handbook of Turbulence, vol. I, Fundamentals and Applications, The University of Tennessee and Space Institute at Tullahoma (Plenum, New York and London, 1977). [7] Y. Kobashi, in: Proc. 2nd Japan Nat. Congr. Appl. Mech. (1952) 223. [8] A.M. Obuchov, Turbulence and the Atmosphere's Dynamics (Gidrometeoizdat, Leningrad, 1988) [in Russian]. [9] G.N. Abramovich, The Theory of a Free Jet and its Applications (CAHI, Moscow, 1936) [in Russian]. [10] G. Comte-Bellot, l~coulement Turbulent Entre Deux Parois Parall~les (Publications scientifiques et techniques du minist~re DF l'air, Paris, 1965). [11] O.V. Troshkin, in: Proc. 2nd sem. Comput. Mech. Computer Design (Moscow-Tashkent, 1988) [in Russian]. [12] O.V. Troshkin, Dokl. Akad. Nauk SSSR 307 (1989) 1072. [13] O.V. Troshkin, To the Wave Theory of Turbulence (Computing Center of Ac. Sci. USSR, 1989) [in Russian]. [14] A.K.M.F. Hussain and W.C. Reynolds, J, Fluid Mech. 41 (1970) 241. [15] A.K.M.F. Hussain and W.C. Reynolds, J. Fluid Mech. 54 (1972) 241. [16] W.C. Reynolds and A.K.M.F. Hussain, J. Fluid Mech. 54 (1972) 263.