About the Prospects for Passage to Instability

The results of the direct numerical integration of the Navier-Stokes equations are evaluated against experimental data for problem on a flow around bluff bodies in an unstable regime. Experiment records several stable medium states for flow past a body. Evolution of each of these states, after losing the stability, inevitably goes by periodic vortex shedding modes. Calculations based on the Navier-Stokes equations satisfactorily reproduced all observed stable medium states. They were, however, incapable of reproducing any of a vortex shedding modes recorded experimentally. The solutions to the classic hydrodynamics equations successfully reach the boundary of instability field. However, classic solutions are unable to cross this boundary. Most likely, the reason for this is the Navier-Stokes equations themselves. The classic hydrodynamics equations directly follow from the Boltzmann equation and naturally contain the error involved in the derivation of classic kinetic equation. Just the Boltzmann hypothesis, which closed kinetic equation, allowed us to construct classic hydrodynamics on only three lower principal hydrodynamic values. The use of the Boltzmann hypothesis excludes higher principal hydrodynamic values from the participation in the formation of classic hydrodynamics equations. The multimoment hydrodynamics equations are constructed using seven lower principal hydrodynamic values. The numerical integration of the multimoment hydrodynamics equations in the problem on flow around a sphere shows that the solutions to these equations cross the boundary and enter the instability field. The boundary crossing is accompanied by appearance of very uncommon acts in scenario of system evolution.


Introduction
The possibility to interpret unstable phenomena becomes very topical.In particular, the reason is conditioned by visualized increase in the intensity of disasters in nature.Experiment expects that the phenomenon of vortex shedding is the example of unstable process.Vortex shedding behind bluff bodies has been studied systematically at least since the days of Strouhal.Vortex shedding is welldefined instability development regime, which is fairly extended along the Reynolds scale.Sometimes, the phenomenon of vortex shedding is called the von Karman instability.Sometimes, the phenomenon is called the Kelvin-Helmholtz instability.However, the study of evolution of solution, after losing the stability, by means of direct numerical integration of the Navier-Stokes equations became feasible comparatively recently.
The phenomenon of vortex shedding behind bluff bodies is taken for evaluation of the results of direct numerical integration of the Navier-Stokes equations against experimental data [1,2].In Section 2, the analysis of numerous divergences between the results of numerical integration of the Navier-Stokes equations and the experiment is given.The analysis is accompanied by corresponding conclusions.In Section 3, the concepts that lie in foundation of the multimoment hydrodynamics [3] are discussed.The characteristic features of unstable solution to the multimoment hydrodynamics equations in the problem on flow around a sphere [4,5] are considered.

Evaluating the Results of Numerical Integration against Experiment
In each of the problems on flow around a bluff bodies experiment records several independent directions of instability development.For 3D flow past a sphere experiment records three stable medium states.The   0 exp s U x stable stationary flow consists of an axisymmetric toroidal recirculating zone in the near wake, which originates a single rectilinear thread in the far wake, see Figure 1(g) in [6].The stable nonaxisymmetric flow consists of two weakly asymmetric halves in the near wake, which originate two rectilinear threads in the far wake, see Figure 12(a) in [7].The x central-type stable state is characterized by periodic restructuring in the near non-axisymmetric wake, which causes wavy motion in the far wake without vortex shedding, see Figure 12(b) in [7].
Each of these three states, , x , and , after losing the stability, starts to evolve in its own direction qualitatively different from the others [2].These directions are schematically shown by three horizontal branches in  After the attainment of a certain critical Reynolds number value 2 , the one-periodic vortex loop street appears in the wake behind a sphere, the upper branch in Figure 1.The periphery of the recirculating zone is periodically detached from the core and moves downstream in the form of hairpin vortex loop.The vortex loops move uniformly along one of the paths of the double undulated thread, which forms the far wake [7].According to experiment, two vortex shedding modes existed at .In the twoperiodic mode, hairpin vortex loops are shed, whereas the 2 two-periodic mode is characterized by vortex ring shedding.As with , the 2 vortex rings are alternately shed from two symmetry related recirculating zone points and rush downstream along different double undulated thread paths [7,8], the upper branch in Figure 1.t V flow loses stability, the middle branch in Figure 1.The periphery of the recirculating zone of the oneperiodic flow separates from its core and moves downstream along one of the double rectilinear thread paths in the form of hairpin vortex loop [9].

 x
After the attainment of a certain critical Reynolds number the toroidal recirculating zone in the near wake behind a sphere begins to pulsate periodically, the lower branch in Figure 1.The frequency of pulsations is anomalously low.The pulsating flow remains axisymmetric [6,10].After the passage of , vortex rings depart from a sphere downstream and move along the spiral path, that is, the For 2D flow past a circular cylinder experiment records two stable medium states.The stable stationary flow is characterized by axisymmetric recirculating zone in the near wake.The centraltype stable state is characterized by periodic restructuring in the near nonaxisymmetric wake without vortex shedding.Experiment finds at least two independent directions of instability development.Vortex shedding along each of the two directions, and [11,12], is characterized by its own characteristic features intrinsic in it.

 
one-periodic flow becomes nonaxisymmetric [10].The attainment of 0 is accompanied by a change in the regime of vortex shedding from a sphere.Vortex rings penetrate into each other and form the 0 twoperiodic continuous spiral sheet in the wake behind a sphere [10], the lower branch in Figure 1.

Re
Evaluating the results of the direct numerical integration of the Navier-Stokes equations against experiment for unstable flows became feasible only recent twenty-I.V. LEBED 216 five years, thanks to the vigorous development of computer facilities.As distinct from the earlier studies, the direct numerical integrations of the Navier-Stokes equations performed recently directly reproduce unstable flow conditions without artificially introducing non-stationary features into the stationary problem.Under artificial modeling a problem with time independent boundary conditions after stability loss is substituted by non-stationary stable problem [13,14].Such substitution is need of initial conditions for non-stationary problem.The initial conditions consist of a solution for a fully developed vortex-shedding flow field.It is necessary to attract experimental data to construct such conditions.In particular, experiment must give eddy sizes and eddy positions in space, initial eddy velocities and vortex shedding frequency, beginning and completion of vortex shedding process at the Reynolds scale.To summarize, such modeling draws a picture of eddies movement.It is incapable of showing the direction of instability development.
Under the linear analysis of stability, the Navier-Stokes equations are linearized for small hydrodynamic values perturbations.Perturbations are expanded in Fourier series in time [15,16].The linear character of studies does not allow evolution calculations to be extended to long times.That is, the linear analysis is incapable of defining the final stable saturated state.However, the linear analysis predicts characteristics of final solution, in particular, the oscillation frequencies around a stable position.
The direct numerical integration of the Navier-Stokes equations gives three stable solutions for flow around a sphere, , , and .These solutions reproduced three stable flows observed experimentally, 0 , , and , correspondently.In all calculations without exception performed by the direct numerical integration of the Navier-Stokes equations, instability development occurs in strict correspondence to classic Landau-Hopf scenario [17].The calculation leads the instability development process in the direction given by dashed slanting line in The bifu a limiting cycle is represented in Figure 2. The Figure 2 was drawn for a flat plate [23].Later, qualitatively analogous pictures were drawn for different bluff bodies [24].The Figure 2 gives the temporal development of the A absolute value of the velocity disturbance amplitu at an isolated point in the wake of a bluff body.After the attainment of the time moment 0 t de  lower solution loses its stability.They are the

U
x solution for a cylinder and the   e contrary, the visualized process of vortex shedding is not limited at time.That is, experiment records a vortex street as long as investigator wants.Moreover, unstable regime, Figure 2, is not a periodic one, because the A disturbance amplitude grows.As a result, time limited non-periodic unstable regime can not be put in correspondence to strictly periodic phenomenon of vortex shedding which has no time frameworks.So, classic hydrodynamics is incapable to come up the expectations of experiment on unstable nature of vortex shedding phenomenon.

The classic calculatio
On th n can put only stable solution in correspondence to vortex shedding phenomenon.As noted above, the classic hydrodynamics equations in problem on flow past a sphere have four solutions, Re Re Re   , but its manifestations are exceedingly indistinct; the intensity of vortex shedding monotonically increases as Re grows.Earlier, this point of view was formulated in [25].An attempt was made [18][19][20][21] [7]).According to Figure 12(c) from [7], the vortex structure appears in the wake suddenly and is fairly well defined starting with the moment of its origination.Secondly, in accordance with calculations, the x solution after the attainment of the Reynolds number value approximately equals to 500 [21].However, in accordance with experiment, periodic vortex shedding is a well-defined mode, which is fairly extended along the Reynolds scale.In the experiments when a sphere was towed through an unperturbed medium vortex shedding was observed over the whole range of Reynolds numbers studied, up to Re = 30000 [10].i.e., calculation predicts the early transition to chaos.This prediction is, however, at variance with experiment.Thirdly, three regimes of the six vortex shedding regimes actually observed are the one-period modes.These include  25 in [20]) give an idea of the total period of recirculating zone oscillations in the x wake behind a sphere after the passage of the * 2 Re critical value.A detailed examination of these flow as executed in [2,20].To summarize, after the arance in the wake behind a sphere, the size of the vortex structure becomes substantial at the surface of the sphere.Subsequently, the structure moves toward the periphery of the recirculating zone, which is accompanied by its continuous dissipation.Lastly, it fully disappears at the periphery of the recirculating zone.
This picture is qualitatively different from the observed full period of oscillation of the recirculat pictures w appe ing zone.As in calculations, the experimental vortex structure engendered begins to expand and move downstream.After reaching the periphery of the recirculating zone, this vortex structure, however, acquires a maximum size rather than dissipates as predicted by calculations.At the end of the period, the vortex localized at the periphery of the recirculating zone separates from this zone.The separation of the shed vortex from the recirculating zone is very clearly shown in Figure 40 in [20].The periphery of the recirculating zone, which is periodically shed from the recirculating zone, rushes downstream and forms a vortex street x (Figure 7(a,i) in [7]).Because of the absence of the detachment of the recirculating zone in Figure 25 in the ca periphery lculation [20], there is no vortex loop street in the wake behind a sphere.This is an essential difference between calculation and experiment.Analyzed Figure 25 from [20], the authors confirmed the disappearance of the vortex structure at the periphery of the recirculating zone.They note that the vortex structure "loses its spiral shape" at the periphery of the recirculating zone.
So, combining of the .The study performed in [28] showed that the flow pictures presented by streaklines, the lines having a form of hairpin vortices appear in near wake behind a sphere.However, a vortex street is absent too in the far wake on flow pictures presented by streaklines.
The authors of all the numerical experiments without exception the wake behind different bluff bodies on flow pictures.Numerical simulations were performed both for 2D and for 3D problems on flow around a circular cylinder and a flat plate, around a sphere and a disk, and so on.However, passed years gave no numerical experiments in which stability loss results in a periodic vortex shedding mode.Then, the conclusion of agreement between calculation and experiment is based in early direct numerical simulations on a comparison of the calculated vorticity distributions with visual observation results [18,21].However, even insignificant twist in a streamline can create closed curves in the vorticity distribution pictures.And, really, the comparison of streamline flow pictures with the vorticity distributions (Figures 25 and 29 in [20]) shows that closed curves in the vorticity pictures correspond to wavy motion in the far wake behind a sphere rather than vortex structures.So, there is no reason at all for identifying the closed vorticity distribution curves in the wake with vortex structures.
That is why, Johnson and Patel [20] do not confine to drawing of streamline curves, str ity distribution curves.To interpret the obvious contradiction between the observed vortex shedding and calculation results they put the regions of the supposed existence of vortex structures


x in correspondence to the observed vortex shedding.The latter direct numerical simulations select name way to interpret the discrepancies between calculation and experiment [26,27].The sense is that both calculated and experimental flow pictures are known to change qualitatively depending on the system of coordinates used to observe or calculate flows.Flow pictures represented by streamlines are not invariant with respect to the Galilean transformation.Vortex structures can be absent in a certain system and well defined in another.It follows that the uniform motion of a coordinate system can mask the vortex structure which, in reality, exists.
Taking this into consideration, Jeong and Hussian developed a method for distinguishing   2 , t x in which vortex structures can hypothetically  exist [28]   2 , t  x regions were indeed capable of predicting the of vortex structures in space.These predictions are, however, fairly approximate.
ly, streamline flow pictures can contain regions with vortex structures that are undetectable by the Jeong-Hussian method.Conversely, this position Name reg method can show the ions of existence for vortex structures where, according to the streamline flow pictures, vortex structures are absent.Moreover, the Jeong-Hussian method outlines the   2 , t  x contours of the supposed existence of vortex structures but does not indicate the particular coordinate system in which these vortex structures are to be sought.
ng-Hussian method cannot be used to correct flow pictures.On the contrary, the correctness of this method is evaluated by comparing the hypothetical regions where vortex structures with calculated flow pictures exist.


x region pattern s not remove the discrepancy between the streamline flow picture calculated in th system of coordinates and the flow picture obser e same system.Really, the experiment records the hairpin vortex shedding in the wake behind a sphere (Fig 7(a,i) in [7]).The streamline flow picture draws the periodic restructuring in the near wake, which causes wavy motion in the far wake without vortex shedding (Figure 25 in [20]).
There may exist such reference system The possibilities of variation the boundary and initial conditions of calculatio bring it in strict correspondence to the experimental boundary and i ns to al conditions ar niti e not very narrowly, the possibilities of perfecting the numerical procedure applied in recent simulations are boundless.Nevertheless, there are almost no prospects for a cardinal change in the calculated flow pictures [2].It is therefore very improbable that the instability development direction calculated in [15,16,[18][19][20][21]  It is also possible that the modes that have been recorded will be differently distributed over the three directions shown in Figure 1, and new modes will force us to modify the concepts upon which the three turbulence development directions are based.However, irrespective of any modifications of the scheme shown in Figure 1, it is very improbable that one of the conclusions drawn from comparing it with experiment will change.Namely, calculations are incapable of reproducing any of the six periodical vortex shedding modes observed along the three turbulence development directions (Figure 1).Most likely [1,2], the reason for simulation failure is the Navier-Stokes equations themselves.

Increase in the Number of Principal
Hydrodynamic Value quations for pair functions and the multimoment ydrodynamics equations are derived from the second of chain of equations, i.e., the BBGKY hierarchy [30].The kinetic e h these equations.The BBGKY hierarchy is closed by the Liouville equation (Figure 3).To pass from the classical mechanics equations to the Liouville equation, the concept of the Gibbs ensemble is invoked.The Gibbs concept is the connecting link between the dynamical deterministic approach and the statistical deterministic approach [30].
The th s equation of the BBGKY hierarchy has the form, Classic hydrodynamics equations exist for abo centuries.By definition, these equations are valid scription of arbitrary continuo to continuity and unlimited deformability principles [29].Statistically grounded hydrodynamics equations are, however, far from being completely established.The greatest progress in this direction was made for one of continuous medium states, namely, for the rarefied gas state, where the characteristic free path  far exceeded the characteristic size of particles d .In a rarefied gas, that is, at d   , the path from classic mechanics equations written separately for each of the medium particle to classic hydrodynamics eq ations was passed without ad l assumptions.The only exception was the Boltzmann hypothesis of molecular s "Stosszahlansatz" (Figure 3).
Classic direction and pair-multimoment direction are two independent branches of the statistical deterministic approach to medium

Clas netics and cl
where is the mass of the particle; is the force of th of the on pa d are the sp rdina ve f th p re-  , , F t x ξ one-particle distribution function obeys he first equation of the BBGKY hierarchy (1).The first equatio losed.The term on the r t d side of (1) that contains the , , , , F t x ξ x ξ two-particle function is responsible for the interaction of particle 1 with some particle 2.
There are several variants of the deriv n directly from the first equation of the BBGKY hierarchy [30].Each of them reaches  2) is valid for a rarefied gas, th (2) at is, at The one-particle distribution function uations (2), (3) has the meaning of th r of particles situated at time in un    1 in e numele-   v ξ ξ (Figure 4).The  ) i At the exit of the region of their n, particles 1 and 2 then have velocities and interactio 2 ξ resp rig 1 ξ ectively (Figure 4), that is, the first term on the ht hand side of (2 s responsible for an increase in the number of 1 ξ -particles caused by collisions.The factorization of two-particle distribution functions in   , , , J t x ξ ξ from Equation (2), that is, their representation in the form of the product of two one-particle functions, The approximation that we use was called the molecular chaos hypothesis "Stosszahlansatz" [31].Classic hydrodynamics equations follow directly from the Bo m x ) ltzann equation (Figure 3) and, naturally, include errors made in the derivation of the classic kinetic equation.Let us elucidate the physical meaning of the error introduced by the Boltzmann hypothesis into hydrodynamics.
Let us pass in Equation ( 2) from 2 F functions written in and 1 1 2 2 , , , The velocity of the center of mass of pair particles, and the modulus of the relative velocity of particles Here , and  and  are the spherical coordinates of the v vector.

Suppose that weight function
, is an arbitrary ies G and v .The properties of the    (8) This means that particle collisions cannot influence the formation of air.In other ords, particle collisions cannot tune the distributions of all these hydrodynamic values to distributions of some other hydrodynamic Th t is, the set of hydrodynamic values constructed on the property of , particle collisions do not participate in the formation of distributions of hydrodynamic values.These distributions are formed under the influence of the initia boundary problem conditions, that is, each hydrodynamic value is principal.As the density of medium increases, the influence of intermolecular collisions on the formation of hydrodynamic value distributions increases.In a continuous medium, L   , this influence becomes predominant.Namely, binary particle collisions tune the distributions of all the function moments to distribution of moments (9).The vel ities G and v are invariants of a particle binary collision.That is why, according to Equation ( 8), moments (9), as previously, do not experience the influence of binary particle collisions.As in the free molecular mode, that is, at remain principal hydrodynamic values.
The equation for the , , f t x ξ one-particle distribution function is written in a si di ensional phase space of one particle ( x-m  space).The dimensionality of the  space allows o perties of particle  , to be accommodated in it; binary particle collision invariants, that is, , , , J t x ξ ξ collision integral strictly correspond to the symmetry proper e Boltz nn collision integral (5), [31].It follows that only three lower partic namely , , J t x ξ collision integral.Let us sequentially accommodate three lower properties of a particle in the ties of th ma le properties,  space.We then have, Equations ( 10) are valid because of symmetry of the partial distribution function with respect to the permutation of th of two pa   2 , , , , F t x G v   e phase coordinates ution functions m (10), [32].
rticles [3].The symmetry properties allow all the moments of the two-particle distrib odd with respect to v to be removed fro It follows that the integrals that contain the lower particle properties , , f t x ξ one-particle distribution function moments, the density of the num ticles   , n t x , medium velocity pressure   , p t x .It follows that, when we pa the rodynamic stage fro e space of one particle, such hydrodynamics equations cannot be constructed using han three lower pr hydrodynamic values co ding to the  space.This passage is closed because Equation ( 2) is not closed.The use of the Boltzmann hypothesis ("Stosszahlansatz") (4) opens up the possibility of approximate passage to hydrodynamics.So, the physical meaning of the error introduced by the Boltzmann hypothesis into hy trary positions of p 2 in space with respect to each ot not closed.There drodynamics is as follows.It follows that just Boltzmann hypothesis allows us to construct hydrodynamics on only three lower principal hydrodynamic values.It follows that the use of the Boltzmann hypothesis excludes higher principal hydrodynamic values (9) from the participation in the formation of classic hydrodynamics equations.Since the classic three-moment hydrodynamics is constructed without the use of values (9), its applicability range is limited to states that are only weakly removed from the statistical equilibrium state.To substantiate this conclusion, we must find passage to hydrodynamics from the phase space capable of accommodateing the whole set of binary particle collision , , , , F t x ξ x ξ two-particle distribution function obeys the second equation of the BBGKY hierarchy (1).The second equation is valid for arbi articles 1 and her.It is are, however, positions of particles 1 and 2 for which the second equation is considerably simplified.
The left hand side of the second equation of the BBGKY hierarchy (1), describes the free movement of particles 1 and 2 and the interaction between the particles 1 and 2. The integral terms containing the , , , , , , , , , , F t x ξ x ξ x ξ ction are responsible for threeparticle distribution fun teraction of particles 1 and 2 with some third par Presuming that a particle may be present at all phase space locations with equal probabilities, introduce the dimensionless quantities, the inticle 3. , In Equation ( 12) the hat appears above the dimensionless quantities, c is the characteristic velocity of the particle, the ratio of x .In this case, the ratio of order of the right hand side terms to order of the left hand side terms of Equation ( 11) is of the order of , d d

 
 .It means that the interaction with third particle has a very weak influenc on the free movement of particle 1 2 n interaction between the particles e s and a d the 1 and 2 at times, proportional to d c .The right hand side terms of Equation (11) receive the fundamental order only at times, proportional to c  .
Let us substitute the force term of the left hand side of the third equation of the BBGKY hierarchy (1) into the the right hand side of Equation (11).Then, , , , , ,  , , , , , ,  F   w ξ ξ .In the right hand side of Equaresponsible for triple collisions convective terms responsible rticles 1 and 2 during the time of 1 with particle 3 are omitted too.
gible with respect to s side of Equation (13) Positions of particles 1 and 3, described by the first and second terms of the right hand side of Equation ( 13), are depicted in  13) is responsible for an increase in the number of 1 ξ -particles caused by collisions with some third The motion and interaction of particles 1 and 2 are, however, described by the left-hand side terms of the second equation of hierarchy (11) that are proportional to , , , , F t x ξ x ξ .For this reason, the terms containing , , , , , F t x ξ x ξ x 1 and 2. The secon must be excluded for particles n of the BBGKY hierarchy (1 e onsiderably simplified, 1) written for particles 1 and 2 (Figure 5), is therefor  .i.e., these particles within a time interval of c  were in the stage of approaching with each other.Thus, the terms of the right hand side of Equatio ( 13) at times, proportional to n c  , descri groups of icles, in which two of three particles (1 and 3) find itself either at the age of approaching before the collision with each r, or in the stage of departure from each other after collision.The second term of the right hand side of Equ ion (11) has the identical physical meaning.
The idea that leads to th oncept of pair distribution functions is as follows.For instance, in a rarefied gas at each time moment, each particle after its last collision moves toward the next collision.This means that every particle 1 in a rarefied gas simultaneously flies away from some particle 2 with which it collided last at point D (Figure 5), and approaches some particle 2 with which it is to collide next at point B.  It follows that it is particle 2 be three part c     st othe at e c plays the role of a third particle 3 for particle 1, with which particle 1 has already experienced a collision and will be in the stage of departure within a time interval not exceeding c  (Figure 5(a)).At the same time, it is particle 1 plays the role of a third particle 3 for particle 2, with which the particle 2 has already experienced a collision and will be in the stage of departure within a time interval not exceeding c  (Figure 5(a)).From the other hand, it is a particle 2 plays the role of a third particle 3 for particle 1, with which particle 1 should collide within a time interval not exceeding c  (Figure 5(b)).At the same time, it is particle 1 plays the role of a third particle 3 for particle 2, , , , , 0 Generally, any m other particle.A medium edium particle forms a pair with every therefore contains   1 N N  pairs of particles.All these pairs are described by the , , , , F t x ξ x ξ function, which obeys the second equation of hierarchy (11).If a single particle 2, which Figu either flies away from ure 5(b)), some pa particle, this function that obey ( re 5(a)), or appr rticle 1, is selected as a pa pair is described by the s Equation (14).N is valid for an arbitrary gas particle rather than some par- , , , ξ x ξ ular particle 1 (Figure 5).For this reason, Equation ( 14) is capable of describing the gas as a whole ous derivation of Equation ( 14) is given in [33].There is n the derivation [33] and Grad method used in deriving the Boltzmann equation for a gas consisting of rigid spheres [30].
The arguments of the , , , , x and 2 x , which are the spa particles 1 and 2. Information about the position of a separate particle in space, however, becomes lost in kinetics and hydrodynamics.Classic kinetics and hydrodynamics deal with the place in space near which a set of particles are situated rather than with particle coordinates.It follows that a necessary condition for the transition from ( 14) to kinetics and hydrodynamics equations is , , , x ξ x ξ vectors to the 12-dimensional space constructed on the x , G ,  , v vectors and produce p distribution functions

 
, , , 15) collects all pairs of particles that either will inevitably collide during the time interval not exceeding the characteristic time between collisions , , , div p f t x G v be velocitie r of pairs of particles approaching (diverging from) each other whose centers of mass are concentrated at time t in a unit volume element near point x and the s of the centers of mass and relative particle velocities, in unit velocity elements near G and v , r - spectively.The relations between pair distribu n functions (15) and , , , , , , f t x ξ from [3] can be used to find t cient (9), Heuristic derivation of a system of kinetic equations for pair distribution functions
A de of the pair tion gi these viscous stress rivation of eq distribu hydrodynami tensors, an uations determ functions is cs equations are  ining lowe ven r moments nd th r th th othe on ( for con ions, to three sep l hydr qu Taking (19) into account, we fi Equations (20b) and (20c) is none othe of the conservation of momentum, and tions (20d), (20e) and (20f) is none tion of conservation of energy.Equati tinuity equation.So, the equation momentum separates into two equat tion for conservation of energy, in tions.In Equation (20), nonprincipa ues and are given by the e at the sum of an the equation e sum o r than the equa-20a) is the conservation of arate equaodynamic val-f Equaand the equa- Here,  is the dynami c viscosity coefficient and s ij U from Equation (21a), the remaining term of the component appears because of an increase in the number of principal hydrodynamic values.The first two components of the q heat flux vector, G q and v q , are principal hydrodynamic values.The first term of the third component, Gv q , is proportional to the temperature gradient from Equation (21b), and the other terms of the third component appear because of an increase in the number of principal hydrodynam c values.
The upper indexes "div" and "app" were omitted in the  [3].The analysis formed in [35] showed that hydrodynamic Equations ( 20), ( 21) could be rigorously reduced to classic hydrodynamics equations only when the state of the system weakly d iated from thermod amic equilibrium.In particular, in the Re x is contained in resulting kinetic equations and hydrodynamics equations [3].The limit of weak non-equilibrium allows us to find the proportionality factor between the   dynamic viscosity coefficient and τ , , The  coefficient can be calculated with arbitrary accuracy by solving the Boltzmann equation with the use of the Chapman-Enscog method for an arbitrary law of interparticle interactions [31].
In calculations of the right hand sides of the equations for t pair functions (18), collision integrals were cal-he cu a gas has exc lated in [3] for consisting of rigid spheres with diameter d .This case eptional advantages, because, in integration in velocities G and v with the  , weight function, the collision integrals of kinetic equations are expressed exactly in terms of the moments of pair distribution functions.The classic Boltzmann collision integral offers similar advantages to the case of so-called Maxwell molecules [36].The ructure of the de ved hydrodynamics Equations (20) a d ( s.The interaction law ly infl ences the values of transfer coefficients.It follows that system (20) and ( 21) is valid for the description of gas flows with an arbitrary law of interaction of structureless particles.
In the derivation of hydrodynamics Equations ( 19)-( 21), only terms linear in τ are retained.At the hydrodynamic description stage, the omitted terms correspond to so-called Barnett accuracy [31,36].In follows that hydrodynamics Equations ( 20) and ( ) prese ted above corres okes accuracy of description.The system of equations determining lower moments of the pair distribution functions (20,21)  Equ ( at g the trajectory of the nter of mass of the pair.This property creates the additional relations between the principal hydrodynamic values.The G U , G q and v q vectors, and the G ij p tensor are multicomponent principal hydrodynamic values.It turned out that the use of the conservation property (17) is accompanied by the appearance of additional relati tween each vector and tensor components.Generally, just these relations must close the system (20,21) of hydrodynamics equations.
However, a problem on flow around a fixed solid sphere d in [4,5] by another manner.In [4], the solution to ation 17) was built.The first integrals of Equation ( 17) for a stationary system are the According to [4], a solution to Equation (17) in the problem with time-independent boundary conditions should be sought in the form of a series of the products of the first integrals,   Relations between the dimensionless coefficients and the dimensional c efficients are given in [4].

Conclusions
x .The integration of the Navier okes equations f -St or the problem of flow past a sphere [15,16,[18][19][20][21]  , incapabl producing instability development process in the direction given by dashed slanting line (Figure 1).The dashed slanting line stable from unstable ones, that is, the dashed line gives the boundary of the instability field.So, solutions to the classic hydrodynamics equations successfully reach the boundary of the instability field.As Reynolds grows, these solutions move along the bound- x recorded experimentally at moderately high Reynolds number values, reaching a few hundreds.This means that the error in stable solutions to the Navier-Stokes equations is not large and cannot distort calculated flow pictures noticeably.This error, however, grows very rapidly after the loss of stability.This is explained by the tendency of nonlinear equations toward causing the divergence of close solutions even in a limited phase space region.This sensitivity to initial conditions was called the Lorentz butterfly effect.The loss of instability development direction is the result of error growth, i.e., the classic hydrodynamics equations become helpless in producing of regular nonstationary periodic flows appearing after the attainment of a certain critical Reynolds number value.
The numerical integration of the multimoment hydrodynamics equations p on a fl a sphere [4,5] shows that the solutions to these equations and enter th stability field.Let in the roblem ow past cross the boundary e in be a stationary   solution of the system (25) and . The retention of three-lower axisymmetric terms in expansion (24) leads to the classic Stokes solution valid in the Re 1  limit [29].For advancement up the Re scale, twenty lower axisymmetric expansion (24) terms are retained in [4].The , stationary solution to the nonlinear algebraic system of twenty equations gives the stationary distribution of the hydrodynamic values (19).According to the   0 , 1, ,20   , solution of the so-called system of the reversed hydrodynamics equations [35].According to [37] infinite.At some point the nonstationary solution is cut off and the urbae break the system finds extremely unlike direction of evolution.It starts a returning motion in the direction of the unstable stationary solution.As a resu he system after losing the stability does not find a new stable position, i.e., after crossing the boundary of the instability field, the evolution of the system takes place only within the instability field.Such a scenario differs from the ideas of classic hydrodynamics, which interprets the devel bifurcations from one stable state to anot e state.
The importance of the entropy in interpretation of instability is revealed in [5].It turns out that the explanation of instability appearance lance.
Namely, an open system with time-independent boundary conditions has a steady state while entropy production in it exceeds entropy outflow through the surface confining the system.Then, any fluctuation generated by the system fades out.As soon as the entropy is removed through the confining surface faster than it is produced in the system, any fluctuation generated by the system begins to grow.The system becomes unstable.
Further studies [38] show that reproduction of the Re Re  , leads to new scenario acts that do not have analogues.Moreover, the   p S t pair entropy of system begins to play the key role to predict the instability development direction.The calculation of the system evolution becomes impossible without participation of the entropy.The multimoment hydrodynamics equations find a large the num r system evolution.And q-2 doi:1 ber of ways fo   p S t pair entropy shows the single way, in which the system evolution moves.
So, the perspectives for removal the discrepancies between calculations and experiment for unstable regimes are revealed along the direction on increase of the principal hydrodynamic values number when deriving the hydrodynamics equations.

Figure 1 .
Instability begins to develop upon the attainment of critical Reynolds number values, and , respectively.Instability development inevitably involves periodic vortex shedding modes.Each of the three turbulence development directions has vortex shedding features of its own only characteristic of the given direction.No matter what direction is selected by experiments, periodic vortex shedding is, however, an unavoidable, well-defined instability-development mode.

Figure 1 .
Figure 1.Three stable medium states originating three turbulence development directions for flow past a sphere.The lower branch corresponds to the evolution of stationary axisymmetric flow   exp s U x 0 Figure 7(a,i) in [7]; and , periodic vortex loop shedding along both double undulated thread branches Re tion, when loses stability, experiences bifu o the

 2 x
regions.The ideas of temporal evolution of vortex structures are based on temporal evolution of boundaries of the the supposed existence of vortex structures represented in Figure31from[20] tends in the w ex behind a sphere to distances much larure ssibly, the flow picture calcu e

1
Ref in which the vortex shedding appears in the calculated flow picture.be identical to the photographs of the flow obtained in the same does not discover such reference system 1 Ref .

Figure 3 .
Figure 3.The scheme of the statistical deterministic approach.


the cylindrical system of coordies with the z axis parallel to the v vector, impa er, and nat b is the ct paramet  is the a uthal angle.That is, the second the right hand side of Equation (2) is responsible for a decrease in the number of 1 ξ rticles in collisions with other particles.function corresponds to a pair of particles 1 and 2, which enter the region of their interaction at velocities  ξ and  ξ , respectively,

Figure 4 .
Figure 4. Interaction region C 0 of a pair of particles with characteristic radius d.If, at the entrance of region C 0 , particles have velocity v and parameters they acquir b ε , , e velocity are invariants of a binary particle collision.The enumeration of all the admissible target parameter valu b and es  and relative motion velocity directions v at fixed G and v values then gives all the possible velocity directions  v .It follows that a collision-caused decrease in the number of pairs of particles from a unit phase volume interval near the 1 , , x G v point characterized by all the adm sible b and is  parameter values and relative motion v orientations is strictly balanced by a collision induced increase in the num r of pairs of particles in this interval with these parameters,  , and integrate the result with respect to velocities.By virtue of Equation (7), we th

 1 , A t x 2 F
is a set of the principal hydrodynamic values, t is local proportionality coefficient.Set (9) contains not only the lower he   function moments but a our anal xi hen lso all the higher moments without exception.To summarize, ysis of the properties of the   1 , , , J t x G v  colli-sion integral reveals the e stence of an infinite number of principal hydrodynamic values.Under free molecular conditions, w L   l and do not fit into the  space.
properties.The transition to the hydrodynamic stage from the phase space of one particle excludes higher principal hydrodynamic values (9) from participation in the construction of hydrodynamics equations.Ho ere is no rigorous passage to hydrodynamics from the

Figure 4 .Figure 4 ,
However, when interpreting the collision integral (13) the vectors v , should be replaced respectively by the vectors w , first term of the right hand side of Equation (

3 ξFigure 5 .ξ
Figure 5. Graphic representation of a pair of particles.withwhich the particle 2 should collide within a time interval not exceeding its atial coordinates within the cylinder of collisions.The 1  and 2  cylinders of collisions are oriented in the direction of the velocity v , the areas of their bases equal the collision cross section area 2 d  , and the heights of the cylinders equal the characteristic free path   , t  x .Let us pass from the 12-dimensional phase space of particles 1 and 2 constructed on the from each other after a collision rs during the time interval not exce The integration with respect to within , v  and  are the spherical coordinates of the v vector.Acco ing to Equation (17), the rd f x vector corresponding to the transfer thermal energy because of movement of th of particles, heat flux vectors corresponding to thermal energy transfer because of relative movement of particles in pairs.Along with does not depend on the form of the law of structureless particle


is not closed.Let us multiply the Equation(17) for the G , and integrate the result with respect to velocities G and v .By virtue of definition of lower moments of pair functions[3], we then have hydrodynamic Equations (20a, b, d, e).Thus, Equation (17) is equivalent to several equations, namely, Equations (20a, b, d, e).According to(17), the

Figure 1
Figure 1 represents observed flow pictures sphere.Experiment records three stable flows:

iС0
i   , solution, an axisymmetric recirculating zone is formed in the wake behind a sphere at Re ~ 20.This recirculating zone has the shape of an axisymmetric toroidal ring.It expands as Re grows but its shape remains unchanged.The calculated distribution t solution is cut off.The reason for i this is analyzed in[4].It turned out that another solution to the nonlinear system of d ations(25) existed in the neighborhood of the cut-off point.It is the

2 -saturation state. Figure 2 was drawn using the data from [23].
2 , Figure 2. The temporal development of the absolute value of the velocity disturbance amplitude A: 0 < t < t 1 -exponential growth; t 1 < t < t 2 -onset of nonlinearity; t > t , the law of large numbers is violated near singular furcations, regions of the coexiste veral stable solutions, e