Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hit-conf.imec.msu.ru/books/NeZaTeGiUs_2010.pdf Äàòà èçìåíåíèÿ: Fri Apr 4 12:06:29 2014 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:14:08 2016 Êîäèðîâêà:

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« » 01 - 07 2010 . . . . «» ­ : . 2010. - 196.

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© . .., 2010


XIX - () « » 1 7 2010 . «» . 60 , 14 . : ( ) , ( , ) . 42 30 . .. .. ­ .. , .. .. « ». . .. , .. , .. ( . ), .. , .. , .. (. ), .. , .. (. ) .. (. --). , 1976. . .. . , 30 «» -

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, . .. (15 2009) .. (25 2008). , .. .. . - , . . . ..-.. ..

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. . 10. .., .., .., .., . . 11. .., .. . 12. .., .., .., .. . 13. .., .. . 14. .. . 15. .., .., .. . 16. .. . 17. .. - . 18. .., .. ­ . 19. .. .

6


20. .. . 21. .., .. . 22. .. . 23. .., .., .., .. . 24. .., .. - . 25. .., .. , . 26. .. . 27. .. - , . 28. .. . 29. .. . 30. .., .., .., .., .. .

7


31. .., .. , . 32. .., .. . 33. .. . 34. .. . 35. .. , . 36. .. . 37. .. . . 38. .. . 39. .., .., .. . 40. .. : . 41. .., .., .. .

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42. .., .. : . . 43. .., .. . 44. .. . 45. .., .., .., .. . 46. .. , . 47. . . . 1. .., .. . 2. .., .. . 3. .., .. 3He. 4. .. .

9


5. .. , . 6. .., .., .. . 7. .., .., .. . 8. .., .., .. . 9. .., .. . 10. .., .., .. - . 11. .., .. . 12. .., .. . 13. .. . 14. .., .., .., .. , . 15. .. .

10


16. .., .., .. . 17. .., .. . 18. .. . 19. .., .., .., .. . 20. .., .. . 21. .. . 22. .. . 23. .., .. . 24. .., .., .. . 25. .., .., .. .

11


26. .. . 27. .. . 28. .. - . 29. .., .. . 30. .., .. . 31. .., .. , . 32. .., .. . 33. .., .., .. . 34. .. . 35. .. .

12


36. .. . 37. .., .. - . 38. .., .. .

13


.. , - . (). , , . -. . , . . . . , () ( ). , . . . , , , . , , . , . . . , . , . , . ,

14


. . () . . , . . , , . . . HYDRODYNAMICAL MODEL OF VACUUM Anatoly Abrashkin Institute of Applied Physics RAS, Nizhny Novgorod The model of a vacuum as a quantum-hydrodynamical continuum is proposed. According to the hypothesis a vacuum consists of virtual particles with Plank's scales named as etherons. Their length, lifetime and mass are expressed in terms of the velocity of light, the gravitational and Plank's co nstants. Particles form the Boze-condensate. Due to the Madelung transformation the SchrÆdinger equation for an e nsemble of non-interacting etherons is reduced to the system of hydrod ynamic equations for an ideal compressible fluid. The density of the fluid is propo rtional to the squared modulus of the wave function. The velocity is propo rtional to the gradient of the phase of the wave function. The dependence of the quantum mechanical pressure versus the fluid density is nonlocal. The equations of dynamics of a vacuum are invariant to the Galilei transfo rmations. From the hydrodynamic point of view they define potential flows only. Orig inal solutions for two -dimensional flows with the quantified circulation and the spherical stationary structures are constructed. These structures are the exa mples of the simplest elementary particles. It is shown that the quantum longitudinal waves of density propagate in a homogeneous vacuum as the gravity waves. According to the general relativity theory the gravity waves are periodical oscillations of metrics. Conventio nally it is assumed that they are similar to the electromagnetic waves, i.e. are transverse, non-dispersive and propagate with the velocity of light. We reject the idea of an absolutely empty space and assume that the gravitational waves represent the density oscillations of vacuum induced by the quantum pressure.

15


As a result, our equation differs from the traditional wave equation. It is of the fourth order and describes the longitudinal waves. The dispersion equ ation for the waves contains the spatial dispersion. The value of their group velocity can't be higher than the velocity of light because the maximal wave number is limited by the inverse Plank's length. The state of the homogeneous and isotropic vacuum turbulence is stu died. The etherons play the role of pulsations. The spatial scale of average tu rbulent fields is much greater than the Plank's length and temporal scale of the fields considerably exceeds the Plank's time. At these scales the density is constant and the pressure is equal to zero. It is shown that the transverse ela stic waves can propagate in such a medium. They are identical to the electr omagnetic waves. The magnetic and electric fields relate to small perturbations of the average vorticity and to the divergence of Reynold's tensor respectiv ely. The velocity of the wave is equal to the velocity of light. Thus, the photon is interpreted as a quasi-particle of a turbulent vacuum. Cosmological applications of the model are discussed as well. It is a ssumed that the dark matter is a real continuum of etherons in the laminar state. The turbulent continuum radiates photons. But due to the turbulent di ffusion the wavelength of the photon acquires the red shift. .., .. , ­ ( , - , , - ) [1]. 1931 . . : . , 1946 . , , , .

16


, . ­ . - -. , [2]. .

. 1.

1. .. . ­ ., , 1969.-183. 2. Tarunin E. L., Alikina O. N. Calculation of heat transfer in Ranque ­ Hilcsh's vortex tube. International journal for n umerical methods in fluids, 2005, 48, P. 107 ­ 113.
TURBULENT REGIME OF THE HYDRODYNAMICS AND HEAT TRANSFER IN THE RANQUE - HILSCH VORTEX TUBE Antiipina N.A., Tarunin E.L. TThe Perm state university (Perm, Russia) Due to features the vortex effect finds practical application in many various areas of technique and manufacture (vortical refrigerating chambers, thermostats and vac uum pumps, driers, separators, elements of instrumentations) [1]. The effect of temper ature division of gases was open by French scientist Ranque in 1931. According to his experiences a twirled stream of a compressed gas was separated in a vortex tube on two: cold and hot. After experiments of German scientist Hilsch (1946) which have confirmed experiences Ranque, the vortex effect beca me object of researches of scientists of different countries.

17


Difficulties of studying the processes occurring in vortex devices consist of h ydrodynamics and heat exchange in them is described as a complex system of nonlinear equations in partial derivatives. The current in real vortex tubes is three-dimensional and turbulent, pressure difference is great and effects of compressibility are essential. For this reason at numerical researches full Navier ­ Stokes' and energy equations were used. One example of current received as a result of calculations at the assumption axial symmetric of a vortex tube, is shown in figure 2. The main feature of the flow is the vortex zone of the recurrent flow, the existence of which was doubted in many papers [2]. Numerical experiences were fulfilled with different values of the parameters of the problem.

Fig. 1. One example of axial symmetric current

References
1. Merkulov .P. Vortex effect and application in a technics of it. ­ Moscow., Machine-building, 1969.-183p. 2. Tarunin E. L., Alikina O. N. Calculation of heat transfer in Ranque ­ Hilcsh's vortex tube. International journal for numerical methods in fluids, 2005, (48), p. 107 ­ 113.

.. . .. . [1] 2010 . ,

18


. , . , . . ParJava (4D) (), . , , . , . 1. .., .., .., .. . . . .. . -: , « ». 2010. 362 . ABOUT MESO-SCALE TURBULENCE IN THE OCEAN-ATMOSPHERE DYNAMICS S..Arsen'yev O.Y. Schmidt's Institute of Physics of the Earth. Moscow This talk presents the book [1] which is publishing in 2010 year [1]. Developed in [1] the theory of meso -scale turbulence is based on the spatial averaging method that leads to asymmetry of the Reynolds's tensor of stresses and necessity to use the balance equations for angular moments of the ave raged flow and meso-scale turbulent eddies. The system of equations is closed by the law of conservation for moment of inertia which is proportional to square of scale of turbulent eddies.

19


When applying the meso-scale theory to physics of atmosphere, the problems of turbulent wakes behind a body are considered including effects of suspensions and intermittency of flows. The geophysical applications include the problems of formation of ocean currents on equator and global oceanic circulation. The problem of the magnetic dynamo of the Earth is also touched upon. Two chapters are devoted to turbulent currents and heat exchange in pipes and channels where application of the meso-scale theory has allowed to completely describe the classical turbulent velocity structures as well as to find out some new effects. Implementation of the Par Java parallel program environment for cluster systems has allowed to receive four -dimensional (4D) numerical solutions of formation and evolution of tornado, which illustrate the kinematics of processes and the adequacy of calculation to natural observations. The book is offered to specialists in geophysics, hydrodynamics, meteorology, and computer implementations, as well as to students and postgraduates. LITERATURE 1. Arsen'yev S.A., Babkin V.A., Gubar' A.Y., Nikolaevskiy V.N. The theory of meso-scale turbulence. The eddies of atmosphere a nd ocean. Ed. by academician G. S. Golitzin. Moscow- Izhevsk: Institute of the computer calculations, Scientific Research Center "Regular and chaotic dynamics". 2010 year. 316 pp. , .. . .. . ­ ( 2 30 ) , . . , ( ) , :

u ( / H ) sec h2 ( / ) , .
P p0 g z g

(1 ) (2 )


gH

sec h 2 ( / ) ,

20


= H (1-n) (4 AL/3A)1/2 ­ (1),(2), AL A ­ , . (1),(2): P ­ , p0 ­ , H ­ , g ­ , - , z ­ , = x + Vt ­ , n ­ , V = (gH)1/2 ­ . . , , . , ( ). , . ON THE THEORY OF SQUALL STORMS AND TORNADOES Arsen'yev S.. O.Y. Schmidt's Institute of Physics of the Earth. Moscow By definition, the squall storm is brief (from 2 to 30 minutes) increase of wind velocity down to hurricane values without rotatio n, which is observed in the zone of severe thunderstorms or on propagation of rapid cyclones carrying cloudiness and rains. In present work the soliton theory of squall storms is constructed. It is shown that the squall storm is dissipative (turbulent) aut osoliton of the wind velocity and pressure in lower troposphere. This auto soliton is exited on the under cloudy temperature inversion by hurricane winds in the middle and upper troposphere

u ( / H ) sec h2 ( / ) , .
P p0 g z g

(1 ) (2)


gH

sec h 2 ( / ) ,

where = H (1-n) (4 AL/3A)1/2 is width of solitons (1),(2), AL and A are coefficients of horizontal and vertical viscosity, correspondingly . In formulas (1),(2) also: P is air pressure nearby Earth's surface, p0 is air pressure on in-

21


version, H is height of inversion, g is acceleration of gravity, is air density, z is vertical coordinate, = x + Vt is progressing coordinate, n is relative roughness on the Earth's surface, V = (gH)1/2 is velocity of soliton motion. The theory corresponds to observations of squall storms and it allows calcula ting of them. If the progressing storm soliton hits into weakly rotating tornado gene thunderstorm supercell, then it stimulates additional collapse of air pressure and increases wind rotation, generating tornado. In present work we discovered analytical formulas, which describe azimuth component of wind v elocity in tornado (depending on radius). Theoretical calculations are compared with observations the wind velocity inside tornado, which was obtained with the help of mobile Doppler's weather radar in USA. ( , ) .. ­ - () () (), -- - . , () - . , , : , , -- , () . : - , , -- , . , - -

22


-- . . , , - . , 60% -- , 5 15 . .. . . ( 80 ) () -- , , , . ( , , ) -- , , ( , ). -- - -- . -- (, ) , , , , . ( ) , , 70­75 . . , Alberto 30% . -- 2 10 . -- .

23


, -- , -- ( ) . -- , , , , .. . , , , , , . , , . , , , . , . , , , , , . «» .. (1), .. (2), .. (2), . . (2) (1) , (2) " . . .. ", , , , / , . -

24


-- , , . , , . ( ) ( ) . -- , . -- , , , , . . , . ( GLOBAL -Field, http://www.iki.rssi.ru). , . , "" . , , , , , , - , . , . , , , , -- , . 07-02000294.

25


.. , , , , , . . , - - , . , . , , . - , . . : , 10 . , , . Solfatara (Campi Flegrei). , 9 . , . ,

26


, 4 . . . , . , , , . , , , . , . , ­ . , , . ( 09-01-92434) « » ( 1959.2009.1). HIGH-TEMPERATURE WATER-CARBON DIOXIDE FLOWS IN POROUS MEDIA Afanasyev A. Institute of mechanics MSU, Moscow Coupled water and carbon dioxide flows in porous media can take place both in natural flows like those forced by volcanic eruptions and in man -made processes like underground carbon dioxide storage or geothermal energy r ecovery. The pressure and temperature in these flows can consider ably exceed their values in water critical point. Today there are no adequate mathematical models that can in aggregate describe both water and water -carbon dioxide mixture properties in sub- and supercritical regions and the dynamics of their flows in such conditions. Thereby the influence of critical p oint on water flows in porous media is not well understood. In the paper cubic equation of state is used to describe water -carbon dioxide mixture in wide range of conditions including critical conditions for

27


mixture. The equation generalizes well known Peng-Robinson equation and can be used to describe properties not only of hydrocarbons but also of ca rbon-dioxide and water. The real mixture properties measurements are used to determine the equation coefficients. Comparison between experimental mea surements and data calculated via the equation of state shows a good agre ement between the data. For example the error in water density calculation is less than 10 percent. There are worked out effective and fast algorithms for phase equilibrium calculation via pressure, enthalpy and mixture composition ­ those thermodynamic variables that are most suitable for near critical flows calculation. The mixture properties simulations are used in research of it flows in p orous media with application to flows in Solfatara volcano (Campi Flegrei). The flows in porous media that take place in the system are forced by ma gmatic chamber located at depth of 9 km. Magma degassing make hot supe rcritical plume of water-carbon dioxide mixture ascend to shallow layers where the fluid from the chamber mixes with cold meteoric water. In the paper the dynamic of high-temperature water plume is studied that is developed because of high-temperature water source located at depth of 4 km. The source sim ulates the mixture flux from the magmatic chamber. The problem is studied in one-dimensional and three-dimensional cases. There is discovered that in one dimensional case there exists a single temporal value when water critical co nditions are reached. At this time intense phase transfer process s tarts and twophase flow region rapidly develops and expands both in upper and lower d irections. In three-dimensional case the water critical temperature and pressure exist permanently after the moment they are reached in the flow. There is di scovered that there exist not a single spatial point but a whole line where the critical conditions are reached. There is shown that as the system tends to steady state there is vapor single-phase flow exactly over the source. The two phase flow zones develop only in peripheral regions where the hot plume interacts with cold meteoric water. The work is supported by Russian foundation for basic research (N 09 01-92434) and grant for leading scientific schools (1959.2009.1).

28


.. , , 10 . , . - . .. . . .. , . .. , - . , . , , , . ( ), , , ( ), .. . . , «» , -

29


(- ). , , , (Voyager 1 2, Hubble Space Telescope, Ulysses, IBEX .). Voyager 1 2 2004 2007 .., , . ( ), . ( , , .) . .., .. . .., , , . . 1.1-1.5 ( 1600 /). ~30 , ~0.5 . [1]. ( ). 200 , ; ~0,4 . . -

30


E/N ( ­ , N ­ ) . , : , , [1, 2]. . - ( . 1 ). , - . , ~6 , . . 08 -08-90003-_.

31


. 1 - : ) , ) .

1. .., .., .. . // , 2008. . 34. . 15. . 75-80. 2. .., .., .. .// , 2009. . 427. 1. . 32-34.

32


. .., .., .. . .. , , , (Pr=0.05) (. [1]). , «Instabilities and Bifurcations in Fluid Dynamics 2009». Pr=9.2 , . R=1, H=0.92 Rx=0.5 [2]. ­ T. Rez. : 1 ( ) 3 ( ). , , ( ) . , . ­ , . , , , , . (Gr = 1.9105T, Mn = 586T, T 0.15K 1.0K , Re x 0 1500) . . [2], . 33


, , [2] ( ) . [2] T=1.0 K, Rez = 0 "false transient" . , [2] . , ­ [3], .

, . (. [4]). 2496.2008.8 ( 09-08-00230).

34


. 1. V. Polezhaev, O. Bessonov, N. Nikitin, S. Nikitin. Three -dimensional stability and direct simulation analysis of the thermal convection in low Prandtl melt of Czochralski model. The Twelfth International Conference in Crystal Growth, Jerusalem, Israel, Abstracts, 1998, 178. 2. N. Crnogorac, H. Wilke, K.A. Cliffe, A.Yu. Gelfgat, E. Kit. Numer ical modelling of instability and supercritical oscillatory states in Czochralski model system of oxide melts. Cryst. Res. Technol., 2008, 43(6), 606 -615. 3. O. Bessonov, V. Brailovskaya, L. Feoktistova, V. Zilberberg. Numer ical simulation of 2D and 3D convectio n in water-soluble crystal growth processes. International Conference "Advanced Problems in Thermal Conve ction", Perm, Russia, Proceedings. Perm, 2004, 325-330. 4. .. , .. , .. . . , 2003, .2, 4 , .63-105. ..1, ..1, ..1, ..2, .3 1 , 2 , , 3 , , () ­ 10 . . -, , . -, , , «» . . , , ). , -

35


, ­ . , . -, - , , , , , . , , . , -. , . , , ; . 07-08-96039. CONVECTIVE FLOWS IN MAGNETIC NANO-SUSPENSIONS A.A. Bozhko.1, A.F. Glukhov.1, G.F. Putin.1, S.A. Suslov.2, T. Tanjala3 1 Perm State University, Perm, Russia 2 Swinburne University of Technology, Melbourne, Australia 3 Lapeenranta University of Technology, Lapeenranta, Finland Magnetic nano-suspensions (MNS) belong to one of the types of nano fluids ­ colloid suspensions of single-domain particles with the average size of 10 nm. Two types of body forces arise in a non -uniformly magnetized nonconducting fluid placed in a magnetic field. First are the driving magnetic forces that can induce convective motion under certain conditions. The second type are the "resistive" forces which arise because of the distortion of a ma gnetic field due to the fluid motion. The magnetization non-uniformity in MNS can be caused by the temperature variation in the medium and by the non uniformity of the volumetric distribution of magnetic particles. Therefore

36


there exist two main mechanisms of magnetic convection: thermal and co ncentration-driven (magnetophoresis, thermophoresis). In addition, one has to take into account gravitational mechanisms of convection that also have two origins: thermal expansion and variation of concentration of a solid phase due to both the thermal diffusion and gravitational sedimentation of particles. It has been shown that convection in MNS is of an oscillatory typ e. Irregular oscillations were observed in one -, two- and tree-dimensional flows arising in a convection loop, spherical cavity and horizontal, inclined and vertical layers, respectively. A spontaneous excitation and decay of convection have been observed and regimes of chaotic localized states and of standing and propagating waves have been detected. The influence of longitudinal and transverse magnetic fields on the mechanical equilibrium and flows of fluid and the heat and mass transfer has been investigated. The interaction of thermo-magnetic and thermo-gravitational flows has been considered in order to determine the evolution of convective patterns. Understanding of the nature of magneto -convection is required for the use of MNS in such applications as energy conversion devices, various sensors, alternative cooling in micro-electronics and micro-gravitation and in crystal growth control. The work was supported by the grant 07 -08-96039 from the Russian Fund for Basic Research. 3 .., .. , , , 3He [1]. 1) 3He . 2) . 3) . 4) . , 3He , . , , 3He -

37


( , , K , .) , ( v / t ) , v , K E [2]. , rotE , E, . 1. Salomaa M.M. & Volovik G.E. Quantized vortices in superfluid 3He. Reviews of Modern Physics 59 (1987) p.533 2. Boldyreva L.B. & Sotina NB. Superfliud Vacuum with Intrinsic D egrees of Freedom. Physics Essays 1992; 5: 510. 3. .. . . 2002 . .. . .. , ( ) . . , , . , . , , [1-3]: dv dv 1 div (1 C ) v C v s 0, 1 C C s s p j ij e i dt dt Re





38


dsvs 9C ( v v s )1 dt 2 s , , ­ , d s / dt . : U ­ , L ­ , 0, 0 ­ . : Re UL 0 / 0 - , , 60 L /(mU ) - C div C v s 0, t
0 s / 0 - .

0 ( [3]). : 1, 2 ­ , ­ , y0 ­ . . [4]. [5] . , , ( > 0.23). , [5], C2<<1, C2 ~ 1 y0 = 0. [5]. , y0 > 0 . max i(k) Re ~ 10. , .

39


0(y). ­ ( -622.2009.1) ( 08 -0100195). 1. Marble, F.E. Dynamics of dusty gases // Ann. Rev. Fluid Mec h 1970, v.2, 397 446. 2. .. // . . . 2008. 6. . 40­53. 3. .. // , 2009, . 429, 4, . 477­480. 4. .. // . . 1961. . 16. . 3. . 171 174. 5. .., .., .. // . 1998. . 24. 5. . 76 ­ 80. STABILITY OF SUSPENSION COUTTE FLOW UNDER THE PRESENCE OF PARTICLE CONCENTRATION GRADIENTS Boronin S.A. Institute of Mechanics of Lomonosov Moscow State University, Moscow We consider the stability of the dispersed flow between two plates mo ving relative to each other (plane Couette flow) taking into account the finite volume fraction of particles. The suspension flow is described within the framework of a modified two -fluid model. It is assumed that the suspension effective viscosity as well as the interphase force depend on the particle vo lume fraction. The study is aimed at analyzing instability mechanism of part icle-laden flows which arise due to a stratification of the concentration of i nclusions. The system of governing equations for the suspension flow in the nondimensional form is as follows [1-3]:

40


div (1 C )v C v s 0,

1

C



dv dv 1 C s s p j ije dt dt Re



i



dsvs 9C ( v v s )1 dt 2 Here, parameters of particulate phase are denoted by index s; is the volume fraction of particles; individual derivative d s / dt is calculated along the particle trajectories. The scales of the flow are as follows: U is the half of the relative velocity of plates; L is the half of the height between the plates; 0, 0 are viscosity and density of the carrier phase. Non-dimensional parameters include: Re UL 0 / 0 is the Reynolds number, 60 L /(mU ) is C div C v s 0, t
0 the particle inertia parameter (inversed Stokes number) , s / 0 is the particle-to-fluid substance density ratio. Main-flow particle concentration profile 0 is specified analytically and corresponds to the particle accumulation in the vicinity of middle -plane of the flow (the same as in the study of a stratified suspension flow in the plane channel [3]). The particle concentration profile is specified by the following non-dimensional parameter: 1, 2 are particle volume fractions in the vicinity of plates and in the vicinity of the flow middle -plane respectively, is the particle concentration gradient, y0 is the width of the zone of uniform particle distribution. After the linearization, the problem of linear stability is reduced to the e igenvalue problem for the fourth-order ordinary differential equation in terms of the magnitude of the stream-function perturbation. The eigenvalues are calculated by means of the orthohonalization method [4]. In the study [5], the stability of plane Couette flow of the dusty gas with the non-uniform particle concentration profile was analyzed. It was shown that the dispersed flow is unstable in the range of Reynolds number values if the magnitude of the particle mass concentration exceeds some threshold va lue ( > 0.23). The problem formulation presented in [5], in the framework of the present study corresponds to the limiting case C2<<1, C2 ~ 1 and y0 = 0. The results of the calculations is in agreement with the study [5]. It is found that in the case of y0 > 0 the growing waves exist even at low Reynolds numbers. Maximal increment of the wave growth max i(k) is reached at Re ~ 10. Numerical calculations show that both particle-to-fluid density ratio and particle inertia parameter do not affect the stability of the suspension flow significantly. Major effect on the stability is produced by the variation of shape of the particle concentration profile 0(y).

41


The work is supported by the grant of the President of Russian Federation for the support of young Russian researchers with Ph.D. degree ( 622.2009.1) and RFBR grant ( 08-01-00195). REFERENCES 1. 2. 3. 4. 5. Marble, F.E. Dynamics of dusty gases // Ann. Rev. Fluid Mech 1970. V.2. P. 397 446. Boronin S.A. Investigation of the stability of a plane -channel suspension flow with account for finite particle volume fraction // Fluid Dynamics. 2008. V. 43. 6. pp. 873­884. Boronin S.A. Hydrodynamic stability of stratified suspension flow in a plane channel // Doklady Physics. 2009. V. 54. 12. P. 477-480. Godunov S.K. On the numerical solution of the boundary-value problem for the solution of the system of linear differential equations // Uspekhi Mat. Nauk [in Russian]. 1961. V. 16. Iss. 3. P. 171 174. Rudyak V.Ya., Isakov E.B., Bord E.G. Instability of the plane Couette flow of two-phase fluids // Letters to ZhTF [in Russian]. 1998. V. 24. 5. P. 76 ­ 80.

.., .. . .. , - . -326 . M = 6, Re1=11.9106 -1. . 500 7. . , . : ) , - 50 42


­ 100 ; ) 30 . . f1, f2 f3 = f1+f2. 0 ( ) 1 ( ). . . , . , . , [1]. ( 50 ) . . , ( ) . , [1]. . , . , . , (. 1) [ 2,3]. , .

43


. , , , , , .

. 1. () (d). f1 + f2 = 300


1. Bountin, D., Shiplyuk, A. & Maslov, A. Evolution of nonlinear pr ocesses in a hypersonic boundary layer on a sharp cone. Journal of Fluid M echanics, 2008 v. 611, p. 427-442. 2. .. , .. , .. , . « - »/ . 2003. . 44. 5. . 64-71. 3. Ndaona Chokani; Dimitry A. Bountin; Alexander N. Shiplyuk; Anatoly A. Maslov Nonlinear Aspects of Hypersonic Boundary-Layer Stability on a Porous Surface// AIAA J. 2005. vol. 43. No.1. P. 149 -155. NONLINEAR PROCESSES IN HYPERSONIC BOUNDARY LAYER AT TRANSITION TO TURBULENCE Bountin D., Maslov A.A. Khristianovich Institute of theoretical and applied mechanics Novosibirsk Investigation of nonlinear processes in a sharp cone boundary layer over solid and porous wall is presented for the weak nonlinear stage of transition to turbulence.

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The experiments are performed in the T -326 hypersonic blowdown wind tunnel of the ITAM SB RAS at a free-stream Mach number = 6 and freestream Reynolds number Re1 = 12106 m­1. The fluctuations of the mass flux are measured by a constant-current hot-wire anemometer. The model is a 7 degree half angle steel cone 0.5 m long, sharp nosed. The model is mounted at zero incidence with an accuracy of 0.1. One longitudinal half of the back part is a solid (roughness is approximately 5m) surface. The other longitudinal half is porous. Two types of porous surface were used: a) regular porosity, which is a perforated sheet, which constitutes the porous surface; the diameter of the holes in the sheet is 50m and the spacing between holes is 100m; the open area (porosity) is 0.2; b) random porosity, which is a felt-metal coating, which is composed of stainless steel fibres of diameter d =30 m; the porosity is 0.75. To reveal nonlinear interactions and their details the bicoherence spectra and statistical analysis are used. The nonlinear phase locking between the three unstable modes can then be detected by measuring the bicoherence spectrum. The bicoherence is bounded by values of zero (no phase locking) and one (absolute phase locking). Statistical analysis allows to detect nonlinear interactions by measuring a deviation of distribution function from normal distribution. The nonlinear interactions in boundary layer over solid wall have been investigated in detail. It has been found that the all main type of nonlinear interactions in the layer of the maximum rms voltage fluctuation involves second-mode disturbances. The most intensive interaction is found to be the subharmonic resonance with detuning. A nonlinear interaction leading to excitation of the second-mode harmonic is shown [1]. Nonlinear processes at the boundary-layer edge in the range of low frequencies (up to 50 kHz) arise far before they appear in the layer of the maximum rms voltage fluctuation. Nonlinear processes in the regions above and below the maximum rms layer are fairly intense even when nonlinear intera ctions (of the quadratic type) have almost disappear from the layer of the maximum rms voltage fluctuation. At the late stages of the transition, the nonlin ear processes reach beyond the boundary layer, forming a turbulent boundary layer [1]. Influence of porous coating on nonlinear interactions has been studied. Nonlinear interactions on the porous wall appear more downstream than for the solid wall. Nonlinear processes in the layer of the maximum rms fluctuation are damped, in particular subharmonic resonance is damped (Fig. 1) [2,3]. Nonlinear interactions are mor e intense out of maximum rms fluctuation

45


layer (lower and over of the layer). Nonlinear processes are delayed on porous surface, but porosity does not change basic mechanisms of the nonli nearity. Such behavior can mean that suharmonic resonance plays cata lytic role in energy transfer from the mean flow to the low frequency disturbances, as it happened in subsonic case.

Fig.1. Bicoherence spectra for porous () and solid wall (d). Nonzero amplitudes along the line f1 + f2 = 300 kHz show intensity of the subharmonic resonance.

1. Bountin, D., Shiplyuk, A. & Maslov, A. Evolution of nonlinear pr ocesses in a hypersonic boundary layer on a sharp cone. Journal of F luid Mechanics, 2008 v. 611, p. 427-442. 2. .. , .. , .. , . « - »/ . 2003. . 44. 5. . 64-71. 3. Ndaona Chokani; Dimitry A. Bountin; Alexander N. Shiplyuk; Anatoly A. Maslov Nonlinear Aspects of Hypersonic Boundary-Layer Stability on a Porous Surface// AIAA J. 2005. vol. 43. No.1. P. 149 -155. .., .., .., .. . .. , -

46


, . , , - . -, , . . , . . , (.1), (.2). -327 = 21, Re1 = 6105 -1 0 = 1200. . - . , . .
0.2

Af0
0.16

0.12

0.08

0.04

0

0

0.4

0.8

1.2

A

1.6

47


. 1
0.01

A

f1

0.008

0.006

0.004

0.002

0

0

0.4

0.8

1.2

A

1.6

.2.

. , : , . . . , , , , . , . , . ( 09 08-00679), ( 2.1.1/3963) 11 ( 9).

48


BISPECTRAL ANALYSIS OF NONLINEAR INTERACTION OF DISTURBANCES IN THE HYPERSONIC SHOCK LAYER OVER THE PLATE Bountin D., Mironov S.G., Poplavskaya T.V., Ciryul nikov I.S. Khristianovich Institute of theoretical and applied mechanics Novosibirsk Results of the complex experimental-calculation research of nonlinear wave processes are presented for hypersonic shock layer on a flat plate with introduction of the artificial disturbances entered into a shock layer from a surface. Calculations are based on the code created in ITAM SB RAS in a frame of the solution of full two -dimensional non-stationary Navie-Stocks equations using shock-capturing method of high order accuracy. Disturbances of periodic blowing-suction type were generated over a model surface near to a leading edge and were modeled by a boundary condition for the cross-section mass flow on some area of a surface of the plate. The behavior of density and pressure pulsations in the shock layer over the plate with change of an angle of attack is studied. The parametrical researches including variations of intensity and position of the source of disturbances are carried out. Values of pulsation amplitudes of density for the basic frequency and the first harmonic depending on amplitude of initial disturbances are shown in figures. It is seen, that nonlinearity on the basic frequency does not exist (fig. 1), but at the same time fast growth of the first harmonic is observed (fig. 2). Experiments were carried out in hypersonic nitrogen wind tunnel T 327A of ITAM SB RAS at a free stream Mach number 21, fixed unit Rey nolds number 6105 m-1 and total temperature 1200 K. To inject disturbances in the shock layer of the model cylindrical oblique-cut gas-dynamic whistle is used. Value of mean density and pulsation of density are measured by means of method of electron-beam fluorescence. High harmonics presence in a spe ctrum of density pulsation is detected and fast growth of the harmonics under model angle of attack increase is detected as well. Good agreement with ca lculation results is obtained. To reveal nonlinear phenomena, experimental and DNS calculation data have been analyzed by bicoherence method. This method allo ws not only to reveal nonlinear processes, but also to determine some characteristics of no nlinear interaction: interaction amplitude and frequencies of waves participa t-

49


ing in interaction. For the first time bicoherence method has been applied for calculation data. For the correct application this method has been i mproved.
0.2

Af0
0.16

0.12

0.08

0.04

0

0

0.4

0.8

1.2

A

1.6

. 1

0.01

A

f1

0.008

0.006

0.004

0.002

0

0

0.4

0.8

1.2

A

1.6

.2. Using bicoherence spectra it is shown that harmonic of injected distur bances in the experiment and calculation grows up due to nonlinear intera ction. Nonlinear interactions not visible in explicit form in pulsation spectra are revealed. In particular, appearance of the second harmonic of injected distur bances is obtained with the help of bicoherence method. The work was supported by the Russian Foundation for Basic Research (Grant 09-08-00679), ADTP (project No.2.1.1/3963) and Program of basic research of Presidium RAS (project No. 9).

50


.., . , , 3D , ­ ( , , ). ­ . , ­ . ( 1 ) . (, - 1 1 ) , . 1. . - . 2. . . . 3. . 4. . . 5. . . 6. . 7. : 4- . 8. . : . 9. ( ) () . . 10. , . . 51


11. , . 2, 4, 5, , . 12. , "" . . 13. . , .. , , , [1]. , , [2]. , , , [3,4,5]. , . [6] , [7]. 10 , . , - . , , . , -

52


, . , . , . , . , « » [8]. : 1. S. Nikiyama The maximum and minimum values of the heat transmitted from metal to boiling water under atmospheric pressure. // Int. J. Heat Mass Transfer, vol.9, 1966, pp. 1419-1433. 2. .. . // .: , 1973. 3. Kobayashi K. Film boiling heat transfer around a sphere in forced-convection. // J. Nucl. Science and Techn., vol.2, 2, 1965. 4. Takehiro Ito, Kaneyasu Nishikawa. Two-phase boundary-layer treatment of forced-convection film boiling. // Int. J. Heat Mass Transfer, Vol.9, 1966. 5. Epstein M., Hauser G.M. Subcooled forced-convection film boiling in the forward stagnation region of a sphere or cylinder. // Int. J. Heat Mass Transfer, Vol.23, 1980. 6. . . . // . , , 12, 1954. 7. M. Van Dyke Perturbation Methods In Fluid Mechanics // Stanford, The Parabolic Press, 1975. 8. Y.Maruyama et al. Unit Sphere Concept for Triggering of Large-scale Vapor Explosions // Nuc. Sci. Technol., 39[8], p. 854-864

53


. . . . . , - , [1] () . [2]: «» . , : . ( ) , , «» , . .

. 1 (,), (. 1). w(, , t ) sin(k0 ) A()eit , w -- . , A () , . A () , , . .





54




, (. 2, ) , , , [1] -- , (. 2, ).

. 2 08-01-00618 -2313.2009.1. 1. .. // . . 2006. 4. . 173-181. 2. .. // . . 2006. . 70. . 2. . 257-263. .., ..,. . .. , - , , , , . . , ( ), . , -

55


. , . , , . . . , , . , , , , . , , , , , , . , . 10 -01-90001 3323.2010.1. A MAGNETIC FLUID BRIDGE BETWEEN CYLINDERS IN THE MAGNETIC FIELD OF A LINE CONDUCTOR Vinogradova A.S., Naletova V.A.,. Turkov V.A Lomonosov Moscow State University, Department of Mechanics and Mathematics, Institute of Mechanics MSU, Moscow The behavior of a magnetic fluid situated between two concentric cyli nders with circular cross section (there is a line conductor in the bulk of the inner cylinder) is investigated. The arbitrariness of wetting angles (0<<) is considered. It is shown that a critical value of the current (or the Langevin's parameter) exists and for currents greater than this critical one there is no bridge between cylinders. It was found that a critical value of the current is greater in the case non-wetting (/2<). It is shown that conditions of the bridge break-up for currents lesser than a critical value have the mainly different appearance in cases of wetting (0<
56


non-wetting the bridge can break up only for the increase of a current in the conductor, and in the case of wetting the break -up can occur in both the increase and the decrease of a current. A drop of a magnetic fluid appears on the line conductor after the bridge break-up. The dependences of the volume of the drop on the drop thickness are calculated for the constant current. It was shown that three critical values of the Langevin's parameter exist, for which the appearance of these dependences changes greatly. It is shown that for the Langevin's parameters greater than the least value of the critical ones both discontinuous and hysteresis changes of the drop thickness can be observed for the change of a current. For the decrease of a current the thickness of the drop on the conductor, which has appeared after the bridge break -up, increases and the bridge rebuilding can occur for the current lesser th an the current of the bridge break-up, or the bridge can not rebuild even for the decrease of a current to zero. The possibility of hysteresis behavior of a fluid shape should be considered for the design of different devices with controlled vo lumes of a magnetic fluid, in which a magnetic field is periodically changed. This study is supported by the Russian Foundation for Basic Research (project 10-01-90001) and the State support of leading scientific schools (project 3323.2010.1). .., .., .. .. , - , , [1] , ( ), . [2], . , . , . , , -

57


. . , . 10-01-90001, -3323.2010.1. 1. .., .., .., .. . .: IX « », -, . - , . 360-364. 2. .., .., .., .., .. . .: 13- , , 2008. . 269-274. VOLUME FORM OF THE HEAVY MAGNETIC FLUID IN A FIELD COIL Volkova T.I., Naletova V.A.,. Turkov V.A Lomonosov Moscow State University, Department of Mechanics and Mathematics, Institute of Mechanics MSU, Moscow In [1] was calculated the dependence of the maximum distance between the horizontal plates, in which the ferrofluid bridge destroyed between them (flat case), of the current in a linear conductor on the top plate. This depen dence is qualitatively coincided with the experimentally measured in [2], which investigated the non-planar magnetic field. In this paper studied the static form of finite volume of heavy magnetic fluid between two horizontal plates in axially symmetric magnetic field coil located on the top plate. One side of the liquid is limited by a cylindrical surface whose axis coincides with the axis of the coil. Was calculated the maximum distance between the plates, in which there is a ferrofluid bridge between them, as a function of current in the coil. It is worth to note that the curve i s nonmonotonic and has one maximum. Specified minimum amount of magnetic liquid needed to create ferrofluid bridge for given parameters of the problem. This study is supported by the Russian Foundation for Basic Research (project 10-01-90001) and the Science School-3323.2010.1.

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REFERENCES 1. Naletova V.A., Volkova T.I., Reks A.G., Turkov V.A. Heavy magne tic fluid between horizontal planes in the magnetic field of a line conductor. IX International scientific conference "Modern problems of electrophysics and electrohydrodynamics liquids", St. Petersburg, Physics Department, St. Petersburg State University, p. 360-364. 2. Naletova V.A., Reks A.G., Savchuk E.L., Taynova A.A., Tsvirko M.I. Stability of magnetic fluid drop in a nonuniform magnetic field. 13th International Conference nanodispersed magnetic fluids, Ples, 2008, p. 269274. . ., . ., . . , , , , , . - . , , . , , ([1], [2]). . - . , . , , . , 08-01-00005- 1. Zaltzman B., Rubinstein I., Electroosmotic slip and electroconvective instability// J. Fluid Mech. 2007. V.579, P.173.

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2. . ., . ., . ., // . 2008. . 421, 4. . 1 -4. ABOUT ELECTROCONVECTIVE INSTABILITY Gusarchuk A. N., Polyanskikh S. V., Demekhin E. A. Kuban State University, Krasnodar The problem of electrohydrodynamic instability arising in the ele ctrolyte solution in the area bounded by a semipermeable membrane due to passage of electric current is under consideration. The importance of the problem is d efined by its numerous applications in micro- and nanofluidics. The problem is described by the ion transport equations, Poisson equation for the electric field, the Stokes equation for creeping flow and the continuity equ ation. Onedimensional solution of the problem wi th no ion advection is found both numerically and asymptotically. This solution is also determined by balance between electro migration and diffusion ([1], [2]). Asymptotical methods are applied due to assumption that the Debye thickness based on the outer characteristic linear scale is small. Numerical solution is found by quasispe ctral taumethod with Chebyshev polynomials as basis functions. Solution o btained is examined for stability to small perturbations, exciting electroconvective movement. The calculations made showed good agreement between numer ical and asymptiotical solution. The research was partially financed by the Russian Foundation for B asic Research grants 08-01-00005-a 3. Zaltzman B., Rubinstein instability // J. Fluid Mech. 2007. 4. E. A. Demekhin, E. M. electroconvection in semipermea 2008. V. 53, P. 450. REFERENCES I., Electroosmotic slip and electr oconvective V.579, P.173. Shapar, and V.V. Lapchenko. Initiation of ble electric membranes // Doklady Physics,
1

60


.., .. -6 . 24- , W = 780 /2, . V = 10, 15, 20, 25, 30, 35 / ( ­ 0.2%). . : , . , ( - ). , Cp . , . . p p =1.2. : T(0), ( ). -

61


4- . "" Cp ( , ). , , , Cp ( 3% - 8%), ( 5%). 1. .. . , 35, . , ., 1975, . 3-35. 2. Dzhalalova M.V. Methods of Parachute Stability Improvement. PIA International Symposium. USA, Houston, Febr., 1997, p.1-7. 3. .., .. . ­ ., . - .-. ., 1969. 4. .. . ­ .: , 1973. NUMERICAL AND EXPERIMENTAL PARACHUTE INVESTIGATIONS WITH DIFFERENT VALUES OF REEFING Dzhalalova M.V., Zubkov A.F. Institute of Mechanics, Moscow State University Experimental investigations of a round parachute model have been ca rried out in the Grand Aerodynamic Wind Tunnel -6 at the Institute of Mechanics of the Moscow State University. This parachute model was d esigned with 24-triangular gores manufactured of sections cut from woven parachute cloth of air-permeability parameter W = 780 l/m2s with a vent and sewed stripe for stabilization on the suspension lines (in the same plane of cutting out), on a short distance from a canopy edge. All loads for each reefed parachute variant were being measured at the confluence point at air flow veloc ities V = 10, 15, 20, 25, 30, 35 m/s (longitudial intensity component of the ini-

62


tial turbulence was 0.2%). The parachute canopy attained different shapes from each other depending on the value of reefing. Stable balance of the parachute in the flow should be specially noted: angle oscillations exceeding a common pulsations were not discovered. Little vibrations of suspension lines and the canopy edge were observed at all values of upstream flow velocities and reefing, but the parachute itself kept a stable position (parachute axis pole-confluence point was parallel to the vector of the airflow). Numerical investigations of the reefing influence on the canopy shape, its aerodynamic drag Cp and the tension T of the radial ribbons were considered. The system of differential equations for the solution of this problem conformably to the considered parachute model was used. This system describes the equilibrium of any radial ribbon element of a round parachute canopy and the point's coordinates on the meridional section of inflated canopy. This problem was solved assuming p = const over the total canopy of parachute. The value of p for the considered parachute model with the known value of air-permeability was assigned by the constant value along the meridional section p = 1.2. The formulated one-parameter boundary value problem has been solved by Newton's method: the tension value T(0) is selected on the pole so that a given value of the line length would be obtained on the canopy edge (the tangent coincides with suspension line on the canopy edge). Runge-Kutta's method of the fourth order precision has been used for a numerical integration of differential equations. The values of the drag coefficient Cp have been obtained at all values of the reefing as a result of the numerical solution of equations taking into a ccount the deduced relation between the values of the reefing and the "imag inary" suspension lines (this relation was being put into boundary cond itions, and then the problem was being solved as if for an usual round par achute). Program for computation of drag coefficient, tension of the radial ribbon and profiles of the tested parachute model has been developed. The results of computation, brought out due to this program for al l values of reefing, are confirmed by experimental data both for Cp (a scattering is within by 3% - 8 %) and for parameters of the canopy shape (a difference does not e xceed 5 %). REFERENCES 1. Rakhmatulin Kh.A. Theory of Axisymmetric Parachute. Proceedings #35, Moscow State University, Moscow, 1975, . 3-35.

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2. Dzhalalova M.V. Methods of Parachute Stability Improvement. PIA International Symposium. USA, Houston, Febr., 1997, p.1-7. 3. Demidovich B.P., Maron I.A. Foundation of Computational Mathematics. Moscow, 1960. 4. Bakhvalov N.S. Numerical Methods. Moscow, 1973. ., .., .. , . , . , . ­ , , ( ) . , . , , , . , (, ) . ­ , . , . , , , ­ .

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ASYMPTOTI DESCRIPTION OF THE DYNAMICS OF LOCALIZED VORTICES AND VORTEX FILAMENTS VIA CANONIC OPERATOR. Dobrokhotov S.Yu., Maslov V.P., Shafarevich. A.I. We consider the Cauchy problem with localized initial data for the multidimensional linear hyperbolic systems with variable coefficients, describing propagation of localized perturbation in fluid. We suggest a new asymptotic representation for solutions of these problems which is the generalization of the canonical operator. It is based also on a simple relationship between fast decaying and fast oscillating solutions and on boundary layer ideas. Our result is the explicit formula which establishes the connection between initial localized perturbations and wave profiles near the wave fronts including the neighborhood of backtracking (focal or turning) and self intersection points. Also we show that the solitary vortices correspond to the the focal points propagated in the phase space. The wave profiles and the structure of vortices are related with a form of initial sources and also with the Lagrangian man ifolds organized by the rays and wave fronts in the phase space. In particular we discuss the influence of such topological characteristics like the Maslov and Morse indices to metamorphosis of the profiles after passing the focal points. We consider also asymptotic solutions for nonlinear N avier-Stokes equations, localized in a small vicinity of a curve and representing a propagating vortex filament. We show, that such solution are connected with topological invariants of two-dimensional divergence-free vector fields. In particular, equations, governing the propagation of the vortex, are defined on a geometric graph ­ set of the trajectories of such a field. - .., .., .. , - , [1-2]. .

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, - , , . ( ) . (, , ) . , , , , , . ( 09 -01-00340, 08-08-00390). 1. .., .., .., Wen -Lung Lu, Ching-Huei Lin. . // . 2007 . 1, .147 -151. 2. M. Z. Dosaev, V. A. Samsonov, Yu. D. Selyutskii, Wen-Lung Lu, Ching-Huei Lin. Bifurcation of operation modes of small wind power stations and optimization of their characteristics. // Mechanics of Solids, Allerton Press, Inc., 2009, vol. 44, no. 2, pp. 214 -221. .., .. ., - . . . , Fr, , , , . Fr . Fr

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, . , . Fr , «» , , . , Fr. Fr . INTERNAL WAVES GENERATION BY A FOUNTAIN IN A STRATIFIED FLUID Druzhinin O.A. and Troitskaya Yu.I. Institute of Applied Physics RAS, N. Novgorod, 603950, Russia The objective of the present paper is to study by direct numerical simul ation (DNS) and theoretical analysis the dynamics of a fountain created by a vertical jet flow penetrating a pycnocline in a density-stratified fluid. A vertical circular, laminar jet flow of neutral buoyancy is considered which prop agates vertically upwards towards the pycnocline level and penetrates a di stance into the layer of lighter fluid. At a certain height the jet fluid stagnates and flows down under gravity around the up -flowing core thus creating a U0 fountain. The DNS results show that if the Froude number Fr (deN 0 D0 fined by U 0 and D0 , the jet flow velocity and diameter at injection, and

N0 ,

the buoyancy frequency in the pycnocline) is small enough (Fr < 2.5) the fountain top, after a transient, remains axisymmetric and steady. However, if Fr is increased (3 < Fr < 5) the fountain top becomes unsteady and oscillates in a circular flapping (CF) mode whereby it retains its shape and moves around the jet central axis. If Fr is increased further (for Fr > 6), the fountain top periodically rises and collapses in a bobbing oscillation mode (or B mode). The development of these two modes is accompanied by the gener ation of different patterns of internal waves (IW) in the pycnocline. The CF mode generates spiral internal waves, whereas the B-mode generates IW

67


packets with a complex spatial distribution. In both cases, the IW fr equency spectrum is characterized by a maximum-amplitude peak whose frequency coincides with the frequency of the fountain-top oscillations and decreases monotonically with increasing Fr. The dependence of the amplitude of the fountain-top oscillations on Fr is well described by a theoretical solution of the Landau-type two-mode-competition model under an assumption of small super-criticality. .. . .. , . - . . . ( ) . , , MEIS-2 (, 2009). . [1]. . , , MAGIA [2, 3].

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1. Crnogorac N., Wilke H., Cliffe K.A., Gelfgat A. Yu., and Kit E. N umerical modeling of instability and supercritical oscillatory states in a Czochralski model system of oxide melts. Crystal Research and Technology, 43 (2008) 606-615. 2. Schwabe D., Zebib A., Sim B-C. Oscillatory thermocapillary convection in open cylindrical annuli. Part 1. Experiments under microgravity. J. Fluid Mech., 491 (2003) 491. 3. Schwabe D., Benz S. Termocapillary flow instabilities in an annulus under microgravity results of experiment Magia. Adv. Space Res., 29 (2002) 629. THREE-DIMENSIONAL STABILITY OF AXISYMMETRIC FLOWS Ermakov M.K. A.Ishlinsky Institute for Problems in Mechanics RAS, Moscow Modeling and linear stability analysis of axisymmetrical convective co nfined flows are full-filled on a base of matrix approach. A steady basic flow is calculated by Newton-Raphson method in matrix form for residuals of finite-difference approximations. Study of stability of disturbances in form of normal modes in azimuthal directions is reduced to solution of genera lized eigen-value problem. Inverse iteration method with shifts is applied for fin ding of eigen values and eigen vectors. Solution of arising linear equations sy stems (for real numbers for calculation of basic state flow and for complex numbers for eigen-value problem solution) is full-filled by conjugate gradient method with preconditioning. Stability of thermocapillary convection in liquid bridges for high -Prandtl flows is studied, in particularly, for the conditions of the spatial e xperiment MEIS-2 (Japan, 2009). The dependencies of critical thermocapillary Reynolds numbers upon Prandtl number of liquid and liquid-bridge aspect ratio are calculated. The results of comparison of fluid flow parameters and critical limits for the benchmark on the Czochralski method [1]. The influence of the depth of annular pool and the pool rotation rate around own axes on the critical limits of thermocapillary flow under horizontal thermal gradient. The n umerically calculated results on critical limits are compared with results of the MAGIA experiment in microgravity and normal gravity cond itions [2, 3].

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REFERENCIES 1. Crnogorac N., Wilke H., Cliffe K.A., Gelfgat A. Yu., and Kit E. N umerical modeling of instability and supercritical oscillatory states in a Cz ochralski model system of oxide melts. Crystal Research and Technology, 43 (2008) 606-615. 2. Schwabe D., Zebib A., Sim B-C. Oscillatory thermocapillary convection in open cylindrical annuli. Part 1. Experiments under microgravity. J. Fluid Mech., 491 (2003) 491. 3. Schwabe D., Benz S. Termocapillary flow instabilities in an annulus under microgravity results of experiment Magia. Adv. Space Res., 29 (2002) 629. - .. , - , . - , , -, . BOUNDS OF GEVREY-SOBOLEV NORMS FOR SOLUTIONS OF EQUIATIONS OF THE HYDRODYNAMIC TYPE V.A. Zheligovsky International institute of earthquake prediction theory and mathematical ge ophysics, Russ. Ac. Sci., Moscow We suggest a new method for construction of bounds of GevreySobolev norms for solutions of equations of the hydrodynamic type by transforming the equation in the Fourier space and introducing the feedback b e-

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tween the norm and its index. As examples, we derive finite-time bounds of Gevrey-Sobolev norms of solutions to the Euler and Burgers equations, and global in time bounds for solutions to the Vogt -regularised Euler equation and the hyperviscosity-regularised Navier-Stokes equation. ­ .., .. . .. , , . . [1], [2]. , . [3], , [4]. , Re1 /Re2, , [5]. : , [6]. , , , ; .

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1. .. . .: . . 1997. 348 . 2. .., .., .. . . , 2000. . 375. 6. . 770-773. 3. Nikitin N. Finite-difference method for solving the Navier-Stokes equations for incompressible fluid in arbitrary orthogonal curvilinear coordinates. J. Comp. Phys., 2006, 217(2), pp.759-781. 4. .. : . . .....-. . . . 2007. 128. 5. .., .. . . 2010..80, . 4, .16-23. 6. .., .. . . . .2008.5..30 -38. . ., . . , , . , . - . . . ; , , ; . QR-. -

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(N=500 -700). [1, 2] . 1. Rossum J.J. Experimental investigation of horizontal liquid films, // Chemical Engineering Science. 1959. V.11, pp. 35-52. 2. Andreussi P., Asali J.C. Hanratty T.J. Initiation of roll waves in gas liquid flows. //Chemical Engineering Journal. 1985 . V. 31, N1, pp 119-126 WIND GENERATED INSTABILITY IN TURBULENT LAYER Zaitseva A.V., Demekhin E. A. Kuban State University, Krasnodar Turbulent horizontal liquid layer, involved in movement by a turb ulent gas flow, and its stability is considered. Such problems have practical application for description of two -phase flows in gas pipelines, where instability determines beginning of slug regime. Bussinesque hypothesis for turb ulent stresses in combination with Van-Driest relation for mixing length are used. Trivial solution for the waveless flow is found. Stability of this s olution to infinitesimal sinusoidal pe rturbations is investigated. Galerkin method is appled in liquid and gas phases: the eigen-functions are expanded in complete system of basic polynomials; additional relations for Galerkin coefficients from BC's at the interface and rigid boundary are added to complete the sy stem. The obtained algebraic eigen-value problem is solved with QRalgorithm. Rapid changing of the velocity profile near the rigid boundary and the interface lead to a large number of basic func tions (N=500 -700) to solve the problem. For the first time a good agreement of theoretical and exper imental [1,2] data of critical parameters is obtained. LITERATURE 1. Rossum J.J. Experimental investigation of horizontal liquid films, // Chemical Engineering Science. 1959. V.11, pp. 35-52. 2. Andreussi P., Asali J.C. Hanratty T.J. Initiation of roll waves in gas liquid flows. //Chemical Engineering Journal. 1985 . V. 31, N1, pp 119-126.

.

73


.. .. 1 1 dx dy 0 2 x 2 y (1 ) - . dx, dy , [1, .53]. 1. , /. (1) 1 1 dx dy 0; 2 x 2 y

1 1 dx dy 0 2 x t t 2 x t t , t t0, dx dy (2) u' v 0 t x y 2 x t 2 y t u' v' . , (2) dx, dy 0 , , x t y t u' v 0 (3) t x y , x t y t . /. .

(t t )

74


2. , , . (4) u u v v dt dx dy - , - . (5 ) u v dt dx dy (4) (5) u u v v (6) t x x y y , (3) (6) (7) u v t x y (7) , 2. (7) ReD=40. 1. ... .- .: , 1973, 847 .

















.. ..
[1] 40. Re = 40 . , (t1 ) (t 0 ) -

75


. t 0 .

(.1) l h / d " ".

l = 0,281, , -

76


. . [1] , , .

[1] - . , , , ,

u' v 0 t x y
u v t x y

(1 ) (2)

u' v' : . (2) ­ . . , , (2) , . [1]. 1. .. . .// , 2009, .16, 3. 2. M.N. Zakharenkov. Thermophysics and Aeromechanics, Vorticity waves in problems of hydrodynamic stability, 2009, DOI 10. 1134/S0869864309030081.

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, . .., .., .., .. , , . , , ( ). ( 100x30 2, 0,5 ). . 30-40 ; . . (t=40500) (M=1,5 -3). , , . - , . , . , (. .1). , 2D . , , -, . -

78


- [1]. ( ) , ( 10 , 0.7 ). (=1.4), (P0 = 25, 75 ; T = 300 ) . , . 08 -08-00903-

. 1 , . 1. .., .., .. / 882 , . 2008, 38 . TURBULIZATION OF BOUNDARY LAYER BEHIND SHOCK WAVE MOVING ALONG PULSE IONIZED SURFACE AREA Znamenskaya I.A., Orlov D.M., Ivanov I.E., Koroteeva . Yu. Moscow State University, Faculty of Physycs. Experimental investigation and numerical simulation of the gas -dynamic flow occurring when the plain shock wave propagates in the channel with the nanosecond surface distributed high-current discharge ("plasma sheet") initi-

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ated on one of its walls are conducted. The instantaneous (co mpared to gasdynamic times) energy input in the thin layer of air at the channel wall leads to the considerable growth of pressure and temperature near the area of the discharge (with 100mm x 30mm size and 0.5mm width). Shock waves and disturbances arise as a consequence. After 30-40 s all the disturbances weaken; relaxation processes in the discharge area continue, and a thin layer of a warm rarefied gas is formed near the surface. The interaction of the shock wave with quasi -2D area of relaxing plasma of the distributed surface discharge is investigated by means of the shado wgraph technique. The shock wave evolution is studied at different moment s of time after discharge (t=40-500 s) and for different Mach numbers (M=1.5 ­ 3.0). Under these conditions the flow is unstable due to relax ation processes and expansion of the warm air layer. The dynamics of a large -scale disturbance moving faster than the plain shock front ­ the so-called precursor ­ is thoroughly explored. The appearance of high vorticity and turbulence is registered in the flow behind the shock, when it is propagating along the non -equilibrium gas region created by the quasi-continuous system of parallel transversal surface discharges. The non-uniformity of the upstream flow is scarcely visualized whereas the turbulent region behind the shock front exhibits a rapid growth (Fig.1). As far as the geometry of the problem allows us to consider twodimensional flow, the shadow images are compared with the results of nume rical 2D simulation. Calculations are based on the time-dependent Reynoldsaveraged Navier­Stokes (RANS) equations and the high-order Godunov's scheme for solving the Riemann problem. The effects of turbulence are d escribed by the - ­ model. The energy deposition is modeled as an instantaneous change in the initial conditions (internal energy) in the area where the surface discharge is initiated (10cm x 0.7 mm). The air ( =1.4) is chosen as a working medium with the initial unperturbed parameters (P0 = 25, 75 Torr, T = 300 ) and boundary conditions coinciding with the experimental data. This work was supported by Russian Foundation for Basic Research (Grant 08-08-00903-).

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Fig. 1 Shadow image of the shock wave front in case when the plasma sheet was initiated on the bottom surface of the channel in front of the shock. REFERENCES 2. Glushko G.S., Ivanov I.E., Kryukov I.A. Computation of turb ulent supersonic flows / Preprint 882 Institute for Problems in Mecha nics RAS, 2008, 38 p. .. . .. , Tu L [1]. , . , , , .

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[2-4]. . k- [5-6], , [7] . ( 09-08-00307).

. 1.
f (Re k ) [7]

1. .. // . . . . . - .: , 1963, .9-82. 2. Zubarev V.M. Comparative analysis of various k-e turbulence models for laminar-turbulent transition // , 601, , 1997. 52.

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3. .., .. // . . , 2008, .20, 8, .87-106. 4. .. . .279 // . , . . ., . 70 . .. . - .: . , 2009. - 416. 5. Myong H.K., Kasagi N. A new proposal for a k- turbulence model and its evaluation. 1st report, development of the model // Trans. Japan Soc. Mech. Eng., B, 1988, v.54, N 507, p.3003 -3009. 6. Myong H.,K., Kasagi N. A new proposal for a k- turbulence model and its evaluation. 2nd report, evaluation of the model // Trans. Japan Soc. Mech. Eng., B, 1988, v.54, N 508, p.3512 -3520. 7. Epik E.Ya. Heat transfer effects in transitions // Engin. Found. Conf., March 10-15, 1996, New York - San Diego, California, 1996, p.1 -47. BOUNDARY LAYER IN HIGH-SPEED TURBULENT INTENSITY FLUID FLOW Zubarev V.M. A. Ishlinsky Institute for Problems in Mechanics RAS, Moscow Influence of free stream turbulence on laminar -turbulent transition is concatenated with turbulence intensity level Tu and value of its scale L [1]. The basic attention is concentrated on a closing problem the averaged bound ary layer equations by means of turbulence models for calculation of regions with Reynolds's low local numbers, the analysis of influence of parameters with high turbulence intensity in a free stream on development of characteri stics of a flow. For research the wall boundary layers various variants of class ical differential turbulence models are considered, allowing to calculate in the continuous manner of areas with laminar, transitive and turbulent modes of a flow, at the high free stream turbulence intensity. The approach for the i mproved description of the existing experimental and theoretical data on trans ition structure in boundary layer in a range fro m small to high values of local Reynolds numbers [2-4] is offered. Influence of scale and degree of free stream turbulence on turbulent characteristics of transition is in details studied by numerical methods. At a flow of a flat plate an incompressible li quid with the high turbulence degree numerical results on k- model [5-6] which have 83


been written down for total kinetic energy dissipation rate , are compared with test experimental data [7] on profiles of speed and intensity of turb ulence. Work is executed with financial support of the pr ogram of the Russian Federal Property Fund (the grant 09-08-00307).

Fig. 1. Comparison of calculations of local longitudinal friction coefficient cf(Re) on plate with known theoreti-cal formulas and the data of experi-ment [7]. LITERATURE 1. Dryden H.L. Transition from laminar to turbulent flow // Turbul. Flow and Heat Transfer. N.-Y.: Princeton Univ. Press, 1959, p.3 -74. 2. Zubarev V.M. Comparative analysis of various k -e turbulence models for laminar-turbulent transition // Inst. Probl. Mech. RAS, Preprint N 601. Moscow, 1997, 52p. 3. .., .. // . . , 2008, .20, 8, .87 -106. 4. .. . .279 // . , . . ., . 70 -

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. .. . - .: . , 2009. - 416. 5. Myong H.K., Kasagi N. A new proposal for a k- turbulence model and its evaluation. 1st report, development of the model // Trans. Japan Soc. Mech. Eng., B, 1988, v.54, N 507, p.3003 -3009. 6. Myong H.,K., Kasagi N. A new proposal for a k- turbulence model and its evaluation. 2nd report, evaluation of the model // Trans. Japan Soc. Mech. Eng., B, 1988, v.54, N 508, p.3512 -3520. 7. Epik E.Ya. Heat transfer effects in transitions // Engin. Found. Conf., March 10-15, 1996, New York - San Diego, California, 1996, p.1 -47. .. . .. , . , .. . , , . , , , . . . , . -

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-, ­ , ­ . , . . 10-01-00109. ... .., .. , , - [1]. . ( ) ( ) . : (, ). . -324 9.18 /c. , , , .

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, , ( 8.37 ) 2380 1000 . . . , . . ( ) 291 . , . 80 8 , . , . 8 12 2 14 . . x = 400Â1200 ( ) 10 21. ( z), x ( 100 ), x z-. U / U e 0.6 , (Ue ­

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). , ( ) , ( ). . , . , . . ( 10-01-00109). 1. A.V. Boiko et al., Steady and unsteady Goertler boundary-layer instability on concave wall, European Journal of Mechanics B/Fluids (2009), doi:10.1016/j.euromechflu.2009.11.001 EXPERIMENTAL STUDY OF EXCITATION OF UNSTEADY GæRTLER VORTICES BY SURFACE NON-UNIFORMITIES Ivanov .V.. Kachanov Y.S.,. Mischenko D.A ITAM SB RAS, Novosibirsk In spite of its great practical importance, the problem of linear nonstationary GÆrtler instability of boundary layers on concave walls has been studied thoroughly only recently in theoretical-experimental work [1]. Due to application of new approaches in this work the applicability of linear stability theory to description of GÆrtler instability was proved. This provides basis for further systematic investigation of excitation of non-stationary GÆrtler vortices by different external (with respect to the boundary layer) perturbations. Among the most probable practical sources for GÆrtler-vortices, onset one may mention freestream non-uniformities and turbulence and nonuniformities of aerodynamical surface (vibrations, roughness). The present experimental study is devoted to investigation of the later receptivity mechanism.

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The experiments were carried out in the low-turbulence wind tunnel T324 of ITAM SB RAS. The measurements were performed based on the hotwire technique at controlled excitation of unsteady GÆrtler vortices by means of a special disturbance source, which simulated surface non -uniformities (vibrations). At the experimental conditions the boundary layer under study was developed over a high-precision experimental model which represented a concave surface with radius of curvature of 8.37 m, streamwise length of 2380 mm and 1000 mm in span. Practically zero streamwise pressure gradient was provided along the model by means of an adjustable wall bump mounted above the model. The desired curvature of the model was provided by set of longitudinal ribs made of duralumin and assembled in a rigid frame. Plexiglas sheet attached to the frame formed the test surface of the model. The described design provided very high accuracy required for the model-surface curvature, at the same time this design allowed to minimize spanwise waviness of the surface. The disturbance source was installed into the model flash with the surface at a distance of 291 mm from the model leading edge. The source represented a set of localized surface vibrators placed in a spanwise row with a fixed step. The membranes of the vibrators (made of latex rubber of about 0.8 mm thick) were driven pneumatically with the help of a special bl ock based on 8 powerful loudspeakers. The block was placed outside of the test section but was connected with the surface vibrators by means of flexible pipes. The loudspeakers were controlled electronically and produce air pressure fluctuations which forced membrane oscillations with amplitudes of several dozen microns. The experiments are performed at excitation of unsteady GÆrtler vortices with spanwise period of 8 and 12 mm and frequency from 2 to 14 Hz. At experimental conditions these parameters correspond approximately to most amplified GÆrtler modes. In the streamwise direction the main measurements were performed in range x = 400Â1200 mm that corresponds to GÆrtler numbers (based on the boundary layer displacement thickness) from 10 to 21. Main set of measurements in each studied regime consisted of a set of hotwire spanwise scans performed at several streamwise locations downstream the disturbance source. During these measurements the hot-wire probe was positioned at a distance from the wall which corresponded to the disturbance amplitude maxima in wall-normal profiles. In addition, normal-towall profiles of disturbance amplitudes and phases were taken in the end of the measurement region and also just above the disturbance source (over the membrane center). The linearity of both the receptivity and the stability

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mechanism under study was checked by repeating some measurements at different amplitudes of the surface vibrations. The shape of the vibrator membrane oscillations was measured thoroughly (with the help of a highaccuracy laser displacement-measurement device) in order to represent the modeled surface non-uniformities in Fourier space required for correct analysis of the receptivity problem under study. The performed measurements have shown that surface vibrations do excite in the boundary layer unsteady GÆrtler vortices. The data obtained allow as to determine experimentally the receptivity coefficients for excitation of GÆrtler modes by aerodynamical surface non -uniformities. The work is supported by RFBR (grant 10-01-00109). 2. A.V. Boiko et al., Steady and unsteady Goertler boundary-layer instability on concave wall, European Journal of Mechanics B/Fluids (2009), doi:10.1016/j.euromechflu.2009.11.001


.., .. . .., . , . , . , . . . . , - [1]. -, , -

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. . - [2]. . , , (1). . PISO [3]. . Re=22000. U=13.3 /. . . . . . 12225 , ,, 244500 . y+. . 1 .e-05.
K ( K U ) t

SGS



K S



, , , , . , . OpenFOAM, [4]. . . .

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. 1) .. , .. . . . 1984. N7. .74-82 2) .., ... . ­ .: . 2008. ­ 368 . 3) Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin et al.: Springer, 2002. ­ 423p. 4) Weller H.G., Tabor G., Jasak H., Fureby C. "A tensorial approach to computational continuum mechanics using object oriented tec hniques", Computers in Physics,1998. v.12, 6. p . 620-631 MODELING OF A TURBULENT MIXING LAYER FORMED AT A REWARD-FACING STEP Kalugin V.T., Strijhak S.V. BMSTU, Moscow This paper describes the development and validation steps of comput ational model for a turbulent flow formed at a reward -facing step. subsonic incompressible gas is considered in the given example. The similar flow arises in various devices, including chambers of combustion engines. The main feature of a flow is a formation of large-scale structures in a mixture layer behind reward-facing step which is used as a flame stabilizer. The expansion of a layer of mixture is defined by dynamics of development of large stru ctures which are formed above on a flow. These large structures increase as a result of merge of the neighbor vortexes and involving in a layer of mi xture of viscous gas from the basic stream. The mixture process in large -scale structures is defined by a small-scale turbulence. The given problem is a good test example, as it allows to spend a pprobation of mathematical model. There are experimental data for a model of the two-dimensional chamber of combustion for a case of an inert and reac ting flow. The data are received by means of LDA measurements and shliren photos [1]. The decision of RANS equations, closed by means of semiempirical model of turbulence, appears inefficient at modeling of flows with non stationary large-scale vortical structures. It is expedient to use Large Eddy Simulation for the current modeling. The large -scale structures pay off by means of integration of the filtered Navier-Stokes equations [2]. The box filter is used for the reception of the filtered equations. The small vortexes, which size does not exceed a step of a settlement grid, are simulated by means of

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Smagorinsky's model and model of one -equation eddy viscosity (1). The final volume method is used for discretisation of the transport equations. The received equations for velocity and pressure are solved by means of iter ative PISO algorithm [3]. The two-dimensional and three-dimensional statements are considered. The calculations are executed for Reynolds number Re=22000. The inlet v elocity of a flow is equal U=13.3 m/s. On an inlet of settlement area on ave rage values casual indignations in the form of random noise are imposed. For the wall, conditions of no-slip impermeable wall are set. On the outlet, the cond itions of continuation of flow or nonreflecting boundary conditions are set. The numerical scheme has the second-order of accuracy. The grid is co nstructed on base of hexahedrons. In case of two -dimensional calculation the grid co ntains 12225 cells, for three-dimensional accordingly 244500 cells. The value analysis of y + is carried out. The method of preconditioned conjugate grad ients is used for the decision of received linear systems. The settl ement step on time is equal 1.e-05.
K ( K U ) t

SGS



K S



As a result of calculation values a component of velocity, pressure, sub-grid kinetic energy, pulsation components, spectral characteristics of a pulsation of pressure are received. The obtained data on distribution crosssection and longitudinal components of velocity, profiles of intensity of turb ulence for longitudinal components of velocity are compared to experimental results. The CFD code is based on OpenFOAM which is the programming tool for problems of continuum mechanics [4]. The animation picture of simulation has strongly pronounced chara cter of a non-steady flow. There is a periodic failure of large-scale vortexes. The given mathematical model can be used for modeling of a spatial flow of bodies with more difficult geometry. LITERATURE.

1) R.W. Pitz, J.W. Daily. Combustion in a turbulent mixing layer
formed at a reward-facing step. AIAA Journal 21-11,1983 p. 1565-1570 2) Volkov K.N. , Emeljanov V.N. Large Eddy Simulation in calcul ations of turbulent flows. ­ : Fizmatlit. 2008. ­ 368 p.

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3) Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin et al.: Springer, 2002. ­ 423p. 4) Weller H.G., Tabor G., Jasak H., Fureby C. "A tensorial approach to computational continuum mechanics using object oriented techniques", Co mputers in Physics,1998. v.12, 6. p. 620 -631
.., . . , k-epsilon . , , , . - , . , . , , . 1. .., .., .. // .49, 2, 2008, .74-78. 2. Kaptsov O.V., Schmidt A.V., Non-invariant solutions of the threedimensional semi-empirical model of the far turbulent wake//arXiv:0912.2890v2 [physics.flu-dyn], 11 p. 3. .., .., .. // .21, 12, 2009, . 137-144.

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REDUCTION OF THREE-DIMENSIONAL MODEL IN THE APPROXIMATION OF THE FAR TURBULENT WAKE TO ONE-DIMENSIONAL PROBLEM Kaptsov O.V., Schmidt A.V. Institute of Computational Modeling SB RAS The three-dimensional standard k-epsilon model of turbulence in the approximation of the far turbulent wake behind a body of revolution in a passive stratified medium is considered. The sought quantities are the kinetic turb ulent energy, kinetic energy dissipation rate, averaged density defect and dens ity fluctuation variance. The full group of transformations admitted by this model is found. The model is reduced to the system of the ordinary di fferential equations due to similarity presentations obtained and B-determining equations method. System of ordinary differential equations satisfying natural boundary conditions was solved numerically. The solutions obtained agree with exper imental data. BIBLIOGRAPHY 1. Kaptsov O.V., Efremov I. A., Schmidt A.V., Self-similar solutions of the second-order model of the far turbulent wake// Journal of Applied Mechanics and Technical Physics, V.49, N2, 2008, p.74-78 2. Kaptsov O.V., Schmidt A.V., Non-invariant solutions of the threedimensional semi-empirical model of the far turbulent wake//arXiv:0912.2890v2 [physics.flu-dyn], 11 p. 3. Efremov I. A., Kaptsov O.V., Chernihk G.G., Semi-similar solutions of two problems of free shear turbulent flows// Mathematical modeling, V.21, N 12, 2009, . 137-144. .. . .. , 1. . , . : ,

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, , . : (, , ) ; . - . (Gr), ( ( ) ). , , , : Pr=0,7 ( ). - . , , ( ). 2. . . , , ( , , ). . , , : 1/5, 2/5 -3/5 , , ( ). , , . , , ( , ) .

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3. . . . - . . , . 1. . ., - , , 1937, . 7, 12, . 1463 -1465; 2. Crane L. J., Thermal convection from a horizontal wire, ZAMP, 1959, vol.10, 5, pp 453-460; 3. Fujii T., Theory of the steady laminar natural convection above a horizontal line heat source, Int. J. Heat Mass Transfer, 1963, vol.6, 7, pp 597-606; 4. .., .., .. , , i i, 1991; 5. .., , , , 25, 1973; 6. .., .., , ., -, 1984; 7. Nikitin N.V., Finite-difference method for incompressible Navier ­ Stokes equations in arbitrary orthogonal curvilinear coordinates // Journal of computational physics, 2006, vol 217, pp 759 -781; 8. Nikitin N.V., Third-order-accurate semi-implicit Runge­Kutta scheme for incompressible Navier­Stokes equations // International journal for numerical methods in fluids, 2006, vol 51, pp 221 -233. 9. ., , ., , 2005 .. , - ,

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- ­ . , , , . , - . , - ( ) . , . , ; , . , , , () - . , . - . 08-01-91951. RESONANT NATURE OF WEAKLY-NONLINEAR STAGES OF TURBULENCE ORIGIN Kachanov Y.S. ITAM, Novosibirsk, Russia At linear stages of turbulence origin in wall-bounded shear layers the initial frequency-wavenumber spectrum of instability modes is of secondary

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significance because of satisfaction of the superposition principle and the possibility of almost complete description of the disturbance development process through evolution of their Fourier components ­ the so-called normal instability modes. In contrast, at nonlinear stages of development the use of the Fourier analysis gets more complicated and the character of the instability-wave evolution depends essentially, in general, on their initial spectrum. The weakly-nonlinear stage of the disturbance evolutions is characterized by the following. The instability waves still exist in the flow and even have wall-normal amplitude and phase profiles, which are very close to eige nfunctions of the linear-stability problem. At the same time, the amplitude growth rates of these perturbations start to deviate radically from linear ones due to interactions of modes of the frequency-wavenumber spectrum (including self-interaction) and beginning of some base -flow distortions. The character of the intermodal interactions depends on spectral content of perturbations. This circumstance makes the problem of description o f nonlinear stages of turbulence origin much more complicated. However, in spite of the difficulties indicated above, a significant progress has been achieved during past years in investigations of physical nature of weakly-nonlinear stages of the turbulence origin occurred in wall shear layers, first of all in boundary layers which are of the most importance for aerodynamical applications. It turned out that there is a very strong mechanism of weakly-nonlinear stages of transition, which dominates over other mechanisms so significantly that it becomes predominant at various very di fferent spectra of initial disturbances and plays prime role even in cases of broadband (continuous) frequency-wavenumber spectra. This is the mechanism of subharmonic-type resonances, which can be efficient in a very wide range of the base-flow and disturbance parameters. The present paper is devoted to description of properties of these interactions and of their role in the turbulence production and in estimation of the laminar -turbulent transition location. This work is supported by RFBR. Grant No 08-01-91951.

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.., .., .., .. . .. (). - . - , , . () , . () - [1]. , f . , , . , 12 , 16 21, ReL = 1.44105, 2.88105 4.32105 0.08 0.5, = -10-80, f=10-80.

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, «-», , - . ­ , - - . - . , -327 21 ReL = 1.44105. () , , , . . , , . . . - . ( 09 08-00557), ( 2.1.1/3963) 11 ( 9). 1. .., .., .. . . 2007. . 19. N 7. . 39-55.

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STABILITY AND CONTROL OF DISTURBANCES IN A HYPERSONIC SHOCK LAYER Kirilovskii S.V., Mironov S.G.,. Poplavskaya T.V, and. Tsyryulnikov I.S Khristianovich Institute of Theoretical and Applied Mechanics SB RAS Novosibirsk State University The flow near the leading edges of a flying vehicle moving with a high velocity in the upper atmospheric layers has the type of a viscous shock layer (VSL). Studying the evolution of disturbances and understanding the mech anisms of VSL instability are necessary conditions for the development of e ffective methods for controlling the laminar -turbulent transition in the hypersonic flow around various vehicles. The paper describes a comprehensive numerical and experimental study of the characteristics of disturbances generated in a hypersonic shock layer on a flat plate by acoustic free-stream perturbations and of disturbances introduced into the VSL from the model surface. A method of active (interference) control of intensity of density fluctuations in the shock layer on a flat plate was proposed, verified by computations, and implemented experimentally. The numerical study is performed by the method of direct numerical simulation (DNS) on the basis of full two -dimensional unsteady NavierStokes equations with the use of high-order shock-capturing schemes [1]. In numerical simulations of the problem of VSL interaction with external acoustic perturbations, the input conditions of the main flow are supplemented with one or several plane monochromatic acoustic waves with an amplitude , frequency f, and propagation angle . Perturbations (periodic blowing/suction) generated on the body surface near the leading edge are modeled by imposing a boundary condition for the transverse mass flow on a certain area of the plate surface. Direct numerical simulations are performed in wide ranges of flow parameters and parameters of interaction of disturbances with the shock layer. In particular, the receptivity of the shock layer to slow-mode and fast-mode acoustic free-stream perturbations is studied at Mach numbers 12, 16, and 21, Reynolds numbers ReL = 1.44105, 2.88105, and 4.32105 with the temperature factor of the surface varied from 0.08 to 0.5, angles of propagation of acoustic waves = -10-80, and frequencies of external perturbations f=1080 kHz. The specific features of interaction of disturbances generated by a blowing/suction source with the VSL are also considered in a wide range of

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the governing parameters, including variations of the amplitudes of the blo wing/suction perturbations and the source location on the plate. Despite a significantly different character of excitation (the acoustic wave affects the entire VSL length, whereas the blowing/suction source is localized near the leading edge of the flat plate), the disturbances arising in the VSL are spatially similar. For this reason, the interference scheme of controlling the intensity of oscillations in the VSL can be realized with an appr opriate choice of the blowing/suction phase and amplitude. The DNS results for some flow conditions are compared with the data measured in the shock layer on a flat plate at a zero angle of attack in a T 327A hypersonic nitrogen wind tunnel based at ITAM SB RAS at the Mach number of 21 and Reynolds number ReL = 1.44105. The possibility of suppression (amplification) of disturbances generated in the hypersonic shock layer on the plate by fast-mode external acoustic waves and by perturbations introduced into the shock layer from the model surface is experimentally demonstrated for the first time. An obliquely cut cylindrical whistle is used to introduce controlled periodic perturbations into the shock layer. The acoustic perturbations in the hypersonic free stream of the wind tunnel are generated by a powerful electric discharge in the settling chamber; the instant of this discharge is determined by a pulse synchronized with pressure fluctuations in the whistle cavity. In this manner, the phases of external and internal co ntrolled periodic perturbations are coupled. The phase difference can be changed by delaying the formation of pulses initiating the discharge with r espect to the signal from the frequency divider. The measurements in the shock layer are performed by the method of electron-beam fluorescence of nitrogen, which does not disturb the flow. This work was supported by the Russian Foundation for Basic Research (Grant No. 09-08-00557), by the Analytical Development Targeted Program (Project No. 2.1.1/3963), and by the Presidium of the Russian Academy of Sciences (Program for Basic Research No. 11/9). REFERENCES
1. Kudryavtsev, A.N., Poplavskaya, T.V., Khotyanovsky, D.V. Application of high-order schemes in modeling unsteady supersonic flows. Mat. Model. 2007, V. 19, No. 7, pp. 39-55.

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.., .., .., .. - , . . - . . . , , . , , . , - , . . . . O-H . . . , , . .

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IMPLICIT COMPUTATIONAL TECHNIQUE OF VISCOUS TURBULENT FLOWS Kozelkov .S., Deryugin Yu.N., Denisova .V., Zelenski D.. Russian Federal Nuclear Center - All-Russian Research Institute of Experimental Physics Institute of Theoretical and Mathematical Physics, Sarov The paper dwells on the implicit computational technique of 2D turbulent lows of viscous compressible gas. The 2D Navier-Stokes equation integration technique is based on three-level difference scheme of the second order of accuracy and Godunov method. The scheme is written in delta form. The implicit operator is factorized in coordinates. One-sided differences are used for derivative approximation describing convective transport with implicit operator according to the eigenvalue of Jacobian matrix. The centered difference approximates the terms describing viscous effects. The number of reconstructions based on linear and parabolic piecewise computational grid cell distributions of parameters satisfying monotony condition is used to increase the accuracy of an approximation of space derivatives in explicit operator. Cell edge flows are found with Riemann problem. The difference equations are solved either with matrix or with scalar runs. The technique capabilties are sown with flow through the blades problems. O-H grid is plotted in terms of block approach with geometrical method and conformal mapping for numerical simulation. The comparison of computational results is given for difference scheme parameters and solution reconstruction algorithms. Given here is comparison of computational results of turbulent flows using different turbulent models and experimental data. The extension of the technique is given for 3D case, as well as computational results of 3D turbulent flows and parallelization method of computations. Gicen here are the results of 3D parallel computations in blade channel. - . ., . . , -- , . .

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, . , . THE COUETTE-TAYLOR PROBLEM FOR PERMEABLE CYLINDERS Kolesov V. V., Romanov M. N.. Southern Federal University, Rostov-on Don. The flows of a viscous fluid contained between two permeable rotating infinite concentric cylinders with a radial through -flow are investigated. The main regime in this problem is the circular Couette flow with nonzero radial velocity field component. It is calculated neutral curves, separating area of stability of the main regime from area of its instability. The methods of the theory of bifurcations and numerical analysis have allowed finding the se condary regimes in a vicinity of the intersection point of the bifurcations producing Taylor vortices and azimuthal waves. , .., . .. , . [1] , ( ) , , . , . , , , , , , , -

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( ) [2] ( ) [3]. . , , , [2] . , 40% . , [4]. , , , . , , , (), (), . , , . , , , , . , , , , , . , . .

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1. Lighthill, M.J., "On Sound Generated Aerodynamically: I. General Theory". Proc. Royal Soc. London, ser. A, Vol. 211, No. 1107, 20 Mar. 1952, pp. 564-587. 2. Kopiev, V.F., Zaitsev, M.Yu., Chernyshev, S.A., Ostrikov, N.N., "Vortex ring input in subsonic jet noise", Int. J. of Aeroacoustics, Vol. 6, No. 4, 2007, pp. 375-405. 3. Tam C.K.W., Burton D.E., "Sound Generated by I nstability Waves of Supersonic Flows," J. Fluid Mech, Vol. 138, 1984, pp. 249 -271. 4. Kopiev, V. F., Chernyshev, S. A., "Vortex ring e igen-oscillation as a source of sound", J. Fluid Mech., Vol. 341, 1997, pp. 19 -57. .. . .. , . . . . ( ) , . . , . . , . , , -

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, , [1]. , . ( ). [2] . 09-08-00390-. 1. .. , .. . // , , 2010, 1 ( ). 2. H. Yamamoto, N. Seki, S. Fukusako. Forced convection heat transfer on a heated bottom surface of cavity with different wall -height // Heat and Mass Transfer. 1983. V. 17. P. 73-83. THE HEAT TRANSFER INTENSIFICATION IN THE CUBIC CAVITIES WITH LAMINAR INCOMING FLOW Krasnopolsky B.I. Institute of mechanics, Lomonosov Moscow State University, Moscow The problem of the flow over the cavities is of interest for a wide range of scientific and engineering applications. The one of the areas of r esearch, that can be applicable in building aerodynamics and technical physics, is the investigation of heat transfer intensification from the bottom of the cav ities. In the present work the results of the numerical heat transfer mode ling in the cubic cavities are presented. The viscous heat-conducting incompressible flow (passive heat transfer) was investigated in the domain of the fragment of the channel with the cavity at the one of the walls. The influence of the di fferent parameters of the laminar incoming flux on heat transfer intensification from the bottom of the cavity is discussed. The influence of the incoming flow parameters (the frequency of the pe rturbations, boundary layer thickness) on the flow stability was investigated for laminar boundary layer Blasius velocity profile with periodic perturbations in the boundary layer thickness at the inlet. The calculations showed that the integral heat flux at the bottom of the cavity depend on the frequency of the

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incoming boundary layer perturbations. The range of the frequencies, that lead to the heat transfer intensification was determined. The further analysis demonstrated, that the frequencies that correspond to these effects, are dete rmined by the stability of the shear layer between the cavity and the m ain channel [1]. The mechanisms that lead to the observed intensification, are of interest. The dependency of the integral heat transfer characteristics at the bottom of the cavity on the velocity of the main channel flow (Reynolds nu mber) was investigated for unperturbed Blasius velocity profile at the inlet. The numer ical dependency of the averaged Nusselt number on the Reynolds number is in a good agreement with the experimental one [2]. The work was supported by Russian Foundation for Basic Research u nder the Grant No. 09-08-00390-a. REFERENCES 1. S.Ya. Gertsenstein, B.I. Krasnopolsky. Influence of the perturbation frequency and boundary layer thickness on heat transfer in flow over the cubic cavity // Fluid Dynamics, 2010 (in press). 2. H. Yamamoto, N. Seki, S. Fukusako. Forced convection heat transfer on a heated bottom surface of cavity with different wall -height // Heat and Mass Transfer. 1983. V. 17. P. 73-83. .., .. - . .. . .. , . , . . . . . 09-08-00390-.

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TRANSITIONAL MODES OF DOUBLE-DIFFUSIVE CONVECTION Kuznetsova D.V., Sibgatullin I.N. Faculty of Mechanics and Mathematics of Moscow State University Institute of Mechanics of Moscow State University Flows of fluid with constant temperature and concentration of admixture on boundaries are considered. Peculiarities of transitional modes are shown in the case when thermal expansion coefficient depends on temperature and changes sign. These types of motions can be seen in natural reservoirs and can be very different from ordinary convection. Stability of periodical solutions and transition to stochastic flows are analyzed in plane case. Stability of three -dimensional structures is studied. Applicability of different methods of averaging is discussed. The work was partially supported by RFFI Grant 09-08-00390-. , .. , , . ( ) , . , . , . , , "" , . , , , . , -

111


, , . . , , . , . . , . . , , ( ) . . , , , . , . . , . .. . .. . - , , -

112


. . , , . , . .. , , . -. [1-2] . ( ) ( ) . [3] -. .

113


, . . , . , , . , , -. . " " ( 2.1.1/1399), ­ ( -622.2009.1) ( 0801-00195). 1. Osiptsov A.N. Lagrangian modeling of dust admixture in gas flows// Astrophysics and Space Science. 2000. V. 274. PP. 377­386. 2. .. // .: . 85- .. . .: . , 2008. . 390­407. 3. Cottet G., Koumoutsakos P. Vortex Method: Theory and Practice. Cambridge. University Press. 2000. 320 p.

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..(1), ..(2), ..(1), ..(1), ..(1) (1) . (2) . -- , -- c , . 3 , . , : -- 15 , -- 32 , -- 25 ( , , ); 200 0.04 ( 40 ). - - , , ; , . , , . , . -- GLOBAL-Field (http://www.iki.rssi.ru), ( 115


). , , , . 16 . , .., .. - .. [1]. : , , ; ( ) ; . , [2] . :

ht q x 0

(1 )

116


2 6q 1 qt hh 5 h 5 x

xxx

q2 4 I 20 h q 0 h 2 I 20 h

(2 )

h, q ­ , t, x ­ , ­ , I 2 ­ , I20 ­ . -. (1) ­ (2) , , , . , [3]. , , , (2) , . [4]. () (2), . . - (1) ­ (2). , [5,6]. ( 09 01 -00595). 1. .., .. // . . . 1949 . . 19. 2, . 105 ­ 120

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2. .. // . . . 1967. 1. . 43 ­ 51. 3. .., .. // . . . 2009. 2. . 18 ­ 32. 4. .., .. // . . . 1. . . 2007. 3. . 49 ­ 56. 5. .., .. // . « » .2, . 1. .: , . 24 ­ 32. 6. .., .. - //. . . 2009 ( ). INFLUENCE OF RIGID SURFACE SHAPING AND NON-NEWTONIAN PROERTIES OF FLUID ON HYDRODYNAMICAL STABILITY AND NONLINEAR WAVES IN CAPILLARY FILMS DRIVEN BY EXTERNAL FORCES Mogilevskiy E.I., Shkadov V. Ya. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University In the work some generalizations of classical problem on viscous film flowing down due to gravity [1] are considered.The following problems are studied: influence of non-uniform external force field acting on the film (flow on curvilinear rotating disk is considered); changes of flow parameter in presence of small corrugations on the rigid surface (local of periodic) ; influence of non- Newtonian fluid properties on characteristics of the flow over plane or corrugated surfaces. Equations derived in [2] describing evolution of film integral characteristics and their generalizations were used as governing equations. For non Newtonian fluid they have the following form:

ht q x 0

(1 )

118


2 6q 1 qt hh 5 h 5 x

xxx

q2 4 I 20 h q 0 h 2 I 20 h

(2 )

here h, q ­ local thickness and flow rate of the film, t, x ­ time and coordinate along the film, ­ similarity parameter, I 2 ­ non-dimensional viscosity coefficient, I20 ­ value of the second invariant of strain tensor calculated on the rigid surface for non-wavy flow. Generalized Orr -Sommerfeld equation was used for linear staility analysis. Equations (1) ­ (2) can be used for the problem of film flow over rota ting curvilinear disk with axisymmetric disturbances. One should take into account that similarity parameter depends on distance between point of o bservation and axis of rotation and disturbances grow due to instability co mpetes with thickness decreasing due to spreading. Values of parameter for flow stabilization are determined. These equation were used to describe flow over plane with corrugation with size compared with film thickness and longitudinal size much greater than transversal. In this case a term depended on rigid surface shape derivatives and proportional to film thickness is added to right-hand part of (2). Stationary and nonlinear wave regimes of flow down are investigated. If non-Newtonian fluid properties are taken into account view of dissip ative term (the last one) in (2) depends on rheology. In the work power -law and Eyring models are considered. Results obtained for spectral problem of instability for generalized Orr­Sommerfeld equation and generalized equations (1) ­ (2) are compared. Linear stability of stationary flow with plane free su rface and solitary and periodical nonlinear waves are studied. The work is supported be RFBR (project 09 -01 -00595). REFERENCES 1. Kapica P.L., Kapica S.P. Wave flow of thin layers of viscous fluid. // Zh. Ekper. Teor. Fiz. 1949. V. 19. P.105 ­ 120 2. Shkadov V. Ya. Wave flow regimes of a thin layer of viscous fluid subject to gravity// Fluid Dynamics. V. 2. 1. P. 29 ­ 34 3. Mogilevskiy E.I., Shkadov V.Ya. Flow of thin film of viscous liquid over rotating curvilinear surfaces // Fluid Dynamics. V . 44. 2009. 2. P. 189201

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4. Mogilevskii E.I., Shkadov V.Ya. Effect of bottom topography on the flow of a non-Newtonian liquid film down an inclined plane// Moscow University Mechanics Bulletin V. 62, 3 2007. P. 76 ­ 83. 5. Shkadov V.Ya., Mogilevskiy E.I. Stability of non -Newtonian fluid film flow on vertical plane// "Contemporary problems of mechanics and mathematics". V. 2 1. P. 18 - 32 6. Mogilevskiy E.I., Shkadov V.Ya. Instability and waves on genera lized Newtonian fluid flow on vertical wall// Fluid dynamics. 2009. In press. .. . .. - , , . , , , , . , , , . , , , , . . . , . , , , , . - - , .

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. , . . 1. .., .., .. .-.:,1981,-224. .., .. . , [1,2] [3]. , , . Re 50. . , [4]. , 100 . . , . . , . 07-08-00255-.

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. 1. , 2006, 406, 6, 749-752 2. . ., .. . Chemphys.edu.ru Pdf/2007-03-05-001.pdf 3. .., .. . , 3, 2009.114-119 4. .., .. . //... 2007..81.1. .136-147 CAVITATION FLOW AND GLOW IN THE DIELECTRIC CAVERN Monakhov A.A., Romashova N.B. Institute of Mechanics, Moscow State University. Experimental results on the flow of weakly conducting liquid in a thin channel with insulating cavities. This work is a continuation of study of the effect of the electrification of cavitation, where it was discovered the glow of the liquid [1,2] and a high degree of electrification in the areas of cavitation [3]. In this paper we show that the luminescence of the liquid occurs when multiple high speeds than the formation of cavitation bubbles. Numbers Re did not exceed 50. Glow is discrete in the form of frequent electricity brea kdowns and is accompanied by electromagnetic interference in the radio with good correlation. This is an important argument confirming the theory of ele ctric glow of cavitation bubbles [4]. Found that the cavitation region are a source of high voltage with a potential of more than 100 kV. With the probe analysis of degree of electrification flow depending on its speed. Found that at low flow velocities in the fluid is present only positive component of electrif ication. When cavitation in the flow of both components are detected. Studies have also shown that the electrification of the tunnel wall and the fluid infl uences the development of cavitation.

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-. .., .. . , (IBM Regatta). - , , , . . , «» «» . , 09-07-00424-. 1. .. - . // . .: - , 1997, .189-197 2. .., .. . // . .17..: , .1330, 2004. 3. .. .., .. . , .15, :, , 4, .7-16, 2007. 4. .., .., ...

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-. . :,- , .9, 2,.395 -400, 2008 .., .., .. , , . . . : , , . . . , . . , . . , 45. . [1] , . 10 -01-90001 -3323.2010.1. 1. .., .. // .: II «- ». -

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. , , 2009, .179-183. MOVEMENT OF A DROP OF A MAGNETIC FLUID ON A HORIZONTAL SUBSTRATE IN THE VARIABLE HOMOGENEOUS MAGNETIC FIELD Naletova V.A., Turkov V.A., Pelevina D.A. Scientific research institute of Mechanics of the MSU Let's consider a behavior of a drop of the magnetic fluid lying on a hor izontal hard substrate, in the homogeneous applied magnetic field. The drop of a magnetic fluid is in the thin elastic capsule. The capsule does not slip on the horizontal substrate. The magnetic field changes in the following way: at first vertical field is turned on then the magnetic field turns clockwise until the angle between the field direction and a horizontal plane reaches a given value , and then the field is switched off. Further process repeats cyclically. When the vertical magnetic field is switched on the drop becomes elo ngated along a vertical axis. Thus the center of gravity rises, and then, when the applied field turns, its center of gravit y moves along horizontal direction. When the magnetic field is switched off the center of gravity of the drop sinks under the action of gravity. In this work a displacement of the center of gravity of a drop for one was calculated. It was supposed that the drop in the magnetic field has the form of the ellipsoid. Dependence of the displacement from an angle is investigated. It is shown this dependence has a unique maximum at some angle smaller than 45 . Cases with various differences of density of a magnetic fluid and surrounding liquid are considered. The obtained results qualitatively agree with the experiments on drop with small surface contact area with substrate described in [1]. This study is supported by the Russian Foundation for Basic Research (project 10-01-90001) and the State support of leading scientific schools (project 3323.2010.1). REFERENCES 1. .., .. . .: II «- ». -

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. , , 2009, .179-183. .., .., .. . [1] , . , ( ). . , , , , . . , . . , . . 3323.2010.1. 1. Vera A. Naletova, Turkov, Sergej A. Kalmykov, in a cylindrical channel. In: (ESMC2009), September 7-1 tugal, p. 124. Klaus Zimmermann, Igor Zeidis, Vladimir A. Dynamics of a prolate magnetizable elastic body 7th EUROMECH Solid Mechanics Conference 1, 2009, Instituto Superior TÈcnico, Lisbon, Po r-

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MODEL OF A PROLATE BODY WITH VISCOELASTIC MAGNETIZABLE POLYMER Naletova V.A., Turkov V.A., Kalmykov S.A. Scientific research institute of Mechanics of the MSU A motion and deformation of a thin pro late cylindrical body with ma gnetizable polymer are considered. Mathematical model of such bodies dyna mics which doesn't take into account viscosity has been obtained in [1]. Here the model of a thin magnetizable rod with viscoelastic material (used Feucht's model) is obtained. Problems of free longitudinal and free transversal oscill ations were solved based on this model using some border conditions. Using analytical solutions of these problems the high -frequency small oscillations missing was shown as it can be shown in viscose liquid. Based on the obtained model the plain problem of the motion of the thin cylindrical rod in cylindrical channel under the action of non -uniform magnetic field is solved. Magnetic field is created by the system of electroma gnetic coils, situated on sides of the channel, which switch on and off with some frequency. Non-linear dependence of magnetization on the magnetic field and friction on the channel sides are taken into account. The calculation program was written and tested using problems of free oscillations. Calculations of body motion were preformed for different field frequencies. This study is supported by the State support of leading scientific schools (project 3323.2010.1). REFERENCES 1. Vera A. Naletova, Klaus Zimmermann, Igor Zeidis, Vladimir A. Turkov, Sergej A. Kalmykov, Dynamics of a prolate magnetizable elastic body in a cylindrical channel. In: 7th EUROMECH Solid Mechanics Conference (ESMC2009), September 7-11, 2009, Instituto Superior TÈcnico, Lisbon, Po rtugal, p. 124. .. , , ­ , (

127


, .) (, , , ..). , , [1], : ) ; ) ; ) , . , , : , , , . , , ( A.Boycott) ­ , , [2]. , , . (Re << 1) . , 50 %. . , . -

128


( ) . , , ­ . . : . , 5% 10-60% . ( 0801-00195). 1. .., .., // , . 35, 7, 2009, . 98 --106. 2. Blanchette F.A. Sedimentation in a Stratified Ambient.// Massach usetts Inst. of Technology, 2003. Ph.D. Diss. 156 p. .. . .. , «» - . . s , z. . 1 3 , « », , , , . - 4000 . , . -

129




. , , . , , , , . , , . , , . , 08 -0100489. .. . .. , , , , . , , «» «», (, , ) . , , , , . . .

130


( ) , . , . , . « ». , «» . . . , . ­ , . (08 -01-00195) « » ( 2.1.1/1399). .. . .. , . , . , -

131


. , , . , . . 20 250 / . , , . , , , , . , , . ( 05-08-18244). . 1. .., .., .. // , 2009, . 426, 2, . 179-182. 2. .., .. . 2337724, 2008.

132


. .. , , . , . , . (, ) . (, , ) , . .. , , , . ( , , ) , . , .

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, , , . , -. , . STABILITY OF ROLLS IN ROTATING MAGNETOCONVECTION O.M.Podvigina International institute of earthquake prediction theory and mathematical geophysics, Russ. Ac. Sci., Moscow We consider the onset of Boussinesq convection in a horizontal layer of electrically conducting fluid rotating about a vertical axis with an imposed vertical magnetic field and rigid electrically insulating horizontal boundaries. The instability of the trivial steady state on increasing the Rayleigh number can be monotonic or oscillatory, depending on paramete r values of the problem (the Chandrasekhar, Taylor, kinematic and magnetic Prandtl numbers) . If monotonic instability occurs, the emerging rolls can bifurcate supercritically or subcritically, in the latter case they are unstable to perturbations of the same periodicity. If the rolls bifurcate supercritically; they can be unstable to perturbations in the form of rolls rotated by any angle (without an imposed magnetic field this is called the KÝppers-Lortz instability). For each of the three conditions we derive equations for respective boundaries in the parameter space, and investigate how the boundaries are modified as parameters of the problem are varied. , .., .. (. ) . , .. . 134


. , , .. . . . ­, [1, 2]. , ­. , , , . e /( R1 R2 ) , ­ , R1 , R2 ­ . , . . : , , . , . , . , , . ( e 0.7 ) , . , , ­ . . .

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, , . 1. .., .., .., .. ­ // , 2008, . 15, 2, . 353-365. 2. .., A.., .. c // , 2008, . 11, 4(36), . 94-104. ON THE MOMENT AND FORCE ACTING ON THE INNER TUBE IN THE FLOW OF NEWTONIAN AND NON-NEWTONIAN FLUIDS IN ECCENTRIC ANNULUS Podryabinkin E.V., Rudyak V.Ya. Baker Hughes Russian Science Center (Novosibirsk) It is considered a laminar fully developed enforced flow in annulus. In a common case axes of cylinders have a displacement, so we deal with eccentric annular channel. Inner cylinder can rotate driven by external torque. Flow of this type can realize in drilling processes, heat exchanger, journal bearings etc. When inner cylinder rotates it is under effect of flow induced force and drag moment. In practice this effects can lead to instability of the system and a ppearance of the whirl motion of inner cylinder. The purpose of this work is studying of these force and moment and their dependence on flow parameters and fluid characteristics. These flows have been modeled numerically by the means of algorithm for solving Navier -Stockes equations based on finite volume method [1, 2]. Flows of Newtonian fluids, Po wer Law fluids, Bingham plastic fluids and Herschel-Bulkley fluid have been studied. It has been shown that moment exerted on inner cylinder grows mon otonically with the eccentricity e /( R1 R2 ) , when ­ a distance between axes of cylinders, R1 , R2 ­ their radiuses. When eccentricity is close to unity momentum grows sharply. It caused by appearance of i nverse eddy flow. For Newtonian fluid flow drag torque is proportiona l to rotation speed of the inner cylinder. For non-Newtonian fluids it depends on flow rate through the ann ulus: as rule it decreases with the growth of the flow rate . 136


The forces exerted on the inner cylinder involve pressure forces and vi scous friction forces. It has been shown that the pressure forces give a domina ting contribution. When eccentricity does not exceed some critical value the force exerted on inner tube pushes it to the channel's wall. When eccentricity is high enough ( e 0.7 for Newtonian fluid) radial component of the force reverses the sign and the inner pipe is pushed away from the wall of outer pipe. The critical value of eccentricity which reverses the sign of radial co mponent of the force grows when power index d ecreases for Power Law fluid or when yield stress increases for Bingham and Herschel -Bulkley fluids. Circumferential component of the force has always the same direction and induce whirl movement of the inner cylinder. Dependence of the force and drag torque on eccentricity has the same shape for all rheological models. Modeling results have been integrated in database and interpolation a lgorithm which allows determining the force and moment exerted on the inner tube has been developed. REFERENCES 3. Rudyak V.Ya., Minakov A.V., Gavrilov A.A., Dekterev A.A. An Application of the new numerical algorithm for solving of the Navier -Stockes equations for physical-pendulum-like viscosimeter modeling // Thermophysics & Aeromechanics, 2008, v. 15(2), p. 353-365. 4. Minakov A. V., Gavrilov A. A., Dekterev A. A. Numerical algorithm for solving spatial problems of hydrodynamics with moving solids and a free surface // Siberian J. Industrial Math., 2008, v. 11, No. 4(36), p.94 -104. .. ( , ) . ( ) , . , , - , ,

137


. . 1 , . .

.1.

«» . .2 , . , , , . , , (.3).

138


.2.

, , , ( x d0 ) . . . ­ . ,

139


, , , .

.3. l : ­ ; ­

. , . u A (1 ) y r 0, A , .. ( ) «» ( ) . , ­ . (u, v, w) , (u, v, w) :

(uv)

(2 )

140


(1) u u , . , x , , . 1925. . , , . . u0 , L ­ , d 0 , um L um (3 ) 6,2 d u0 0 ( (3) ). u (r ) , r , ­ . ( u um ) , , : 2 u f ( ) 1 - 3 2 , (4 ) r um , (4) . , , , ( !) , . 3. um ()





141


). , 1925 , «» : u u u , ( u) 2 u y y y , ­ . ­ : 1 K (u2 2 w2 ), 2 u : u K C (5 ) r max ,

u ( u) C K (6 ) r K (.4). , C (5) (6)
:

(uv) 2 K C 0,32 0,1; K (uv)

2

(7 )

C (7), K (5) (6) K . , L d0 , (.5).

142


. 4

.5. : , -

1. , , , . . 143


2. , (, ). 1. .., .., .., , , 2007, 43, 3, . 378-383. 2. .. /. / ../, ., , 1985, 240. 3. .. ., , ., , 1984, 710. 4. .., , , - .-.. ., 1963. 680 . 5. .., .., .., , ., , 19 77 ., 248 . .. ( , ) 1. . , ­ , - . P. , [1]: 0 ( S 0,5 /) , 0 S 1 /), . .. 25 Tad 144


. . 1 . P 0,1 , Tad 0 S . 0 S Tad [6] :
0 S 9,92 103 exp (23000 / Tad )

(1 )

.1. Tad P0 1

, 0 S , 2,15

Tad 3000K , Tad . , , . (OH) . « », - 1938 [5], , , ­ P T:

S AP

0

n 1 2

B exp T m

(2 )

145


n ­ «» , , Tm ­ . 0 n 2 S 0 , n 1 S ~

. P , .., «m» (-0,3) (+0,25) 0 S ( P) ~ P m . ( ), . [4] , : ­ : L 3,95 Re0,5 (d0 d ) 0,5 (3 ) 0 d0 ­ : L 34 Re0, 2 (d 0 d ) 0,5 (4 ) 0 d0 (d0 d ) . , : d 0 , [4]. , , d 0 d 0

1

d 3 103 , ( [4] ) . [4] (3) (4) , ­ S B,T ­ S
B, L

, Re0 (d )

(d0 d ) :

146


S S

B ,T B, L

0,09 Re0,3 (d ) 0

(d0 d )

0,3

(5 )

, (5) , .

(3) (4), , -

S

B, L

, ,

S

B ,T

,

.. . . , 1948 [3], , (5): S B,T 0 (6 ) 0,17 Re0, 24 d 0 , 26 0 S B, L d 0 «», . 2.

.2. Re d 0 S
B ,T

: I - 2

2,

II - 2

4

III ­ 3

8

147


S B , , (. 3). , Re0 103 , .. . S B , .. ( ), . - ([1] ­[2]). , , . , . , , W (T ),

.3. Re0

T. , T2 T0 300 K ,

148


0 S B S , (.4), .

.4.
0 S .

, , , . u0 , , , (5) (6). , , , (1800Â2800 /) (0,4Â3,0 /). 1. ., ., , . . .// .. .. / .: . , 1968, 592 . 2. .. . .: , 1965, 740 . 3. Bollinger L., M., Williams D.T., NACA, TN, 1707, 1948. 4. / / ../, , 2006. ­ 352 . 5. .., - .. // . , 1938, 19, . 693.

149


. .. , . .. , , ( , ­ ), , , . , 70 , - [1], 80 [2]. ( , ) [3]. , - , , , , . , . . - , . , , . ( -

150


, .). , . . ( , ), , . 2496.2008.8 ( 09-01-00230). 1. .. , . , , 4, 1972, 77-88. 2 .., .., .. . . : , 1991, 240 c. 3. .., .., .. . .: . . . -. ., 1989, 320c. . ., . . , , . . - [1,2], . , . , -. ,

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. [3], , . , - . , 08 -01-00005-. . 1. Zeleny J. The electrical discharge from liquid points, and a hydrostatic method of measuring the electric intensity at their surfaces// The Phys. Rev. 1914. V. 3. 2, P. 69-91. 2. Gonzales A., Ramos A., et. al. Fluid flow induced by nonuniform AC electric fields in electrolytes on microelectrodes. II. A linear double- layer analisys // The Phys. Rev. E. 2003. V. 61. 4. P. 4019. 3. Polyanskikh S. V., Demekhin E. A. Stability of non -axisymmetric electrolyte jet in high-frequency ac electric field // Microgravity Sci. Tec hnol. 2009. V. 21. P. S325-S329. STABILITY OF LIQUID ELECTROLYTE JET IN AC ELECTRIC FIELD OF ARBITRARY FREQUENCY Polyanskikh S. V., Demekhin E. A. Kuban State Univercity, Krasnodar The problem of the liquid micro-jet behavior placed in external tangential ac electric field is under consideration. The fluid is supposed to be viscous Newtonian. From the physical point of view liquid is supposed to be ionic conductor, i. e. electrolyte. Electrolytes are the least studied liquids, despite the fact that they were investigated in experiments and often used in practice [1,2]. The corresponding theory still does not exist. In the present work linear stability analysis is investigated theoretically. The linear stability problem arises in the form of eigenvalue problem fo r linear partial differential equation with spatial-dependent time-periodic coefficients. It is shown that monodromy operator has the only real unstable mult iplier among countable set of other eigenvalues. One of the most simple and effective asymptotic me thod ­ the averaging method - is applied to study vibrations with high frequencies [3]. It is ju stified

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by the results obtained by Floquet theory. Averaging method yields an explicit dispersion relation of the problem suitable for high-frequency oscillations of external electric field with a high degree of accuracy. The research was partially financed by the Russian Foundation for B asic Research grants 08-01-00005-a REFERENCES 1. Zeleny J. The electrical discharge from liquid points, and a hydrostatic method of measuring the electric intensity at their surfaces// The Phys. Rev. 1914. V. 3. 2, P. 69-91. 2. Gonzales A., Ramos A., et. al. Fluid flow induced by nonuniform AC electric fields in electrolytes on microelectrodes. II. A linear double -layer analisys // The Phys. Rev. E. 2003. V. 61. 4. P. 4019. 3. Polyanskikh S. V., Demekhin E. A. Stability of non -axisymmetric electrolyte jet in high-frequency ac electric field // Microgravity Sci. Technol. 2009. V. 21. P. S325-S329. .., .. . .. , - : , , . . , , , , , , , .

153


(=21) , . , - [1]. - [2], , . , : (.1), (.2).

. 1
. - - : a ­ , - A=0.03; - A=0.6

. - ­ .

154



2

1 2

1 0 0 0.2 0.4 3

A

.2. x=0.5: 1 ­ , 2 ­ , 3 ­

. . ( 09 08-00679), ( 2.1.1/3963) 11 ( 9).
1. .., .., .. . . 2007. . 19. N 7. . 39-55. 2. . ., . ., . ., . . . . , . 2004. 2. . 16-23.

DIRECT NUMERICAL SIMULATION OF NONLINEAR WAVE PROCESSES IN A HYPERSONIC SHOCK LAYER Poplavskaya T.V. and Tsyryulnikov I.S. Khristianovich Institute of Theoretical and Applied Mechanics SB RAS Novosibirsk State University The analysis of the laminar-turbulent transition is traditionally divided into three stages: problems of receptivity, exponential growth of small pertu rbations, and nonlinear interaction of disturbances with stochastization leading

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to the transition to turbulence. The first and second stages of the laminar turbulent transition have been successfully studied by means of the linear th eory of hydrodynamic stability. In the case of a hypersonic shock layer, ho wever, where the flow is not parallel, the flow is essentially divergent, there is a streamwise pressure gradient, the bow shock wave is located extremely close to the boundary layer, and instability waves can be excited not only by the receptivity mechanism but also by means of direct amplification of distur bances passing through the shock wave, it seems promising to use direct n umerical simulations (DNS) for all stages of the transition, from receptivity to nonlinear interaction. The growth of disturbances excited by external acoustic disturbances on the plate surface in a hypersonic (=21) shock layer on a flat plate is studied by the method of direct numerical simulations. The numerical study is pe rformed with the use of codes developed at ITAM SB RAS, based on solving full two-dimensional unsteady Navier-Stokes equations by means of highorder shock-capturing schemes [1]. The DNS results for external acoustic waves with small initial ampl itudes are in good agreement with the data calculated by the loc ally parallel linear stability theory with allowance for the shock wave influence [2], where the usual asymptotic boundary conditions for disturbances are replaced by the conditions on the shock wave, and with the data of wind -tunnel experiments. The calculations at high initial amplitudes show that nonlinear processes b ecome more essential in the shock layer: distortion of the mean flow (Fig. 1), nonlinear saturation of the amplitude of the first harmonic and rapid growth of the amplitude of the second har monic (Fig. 2).

. 1
. - - : a ­ , - A=0.03; - A=0.6

156



2

1 2

1 0 0 0.2 0.4 3

A

.2. x=0.5: 1 ­ , 2 ­ , 3 ­

The nonlinear evolution of disturbances in a hypersonic shock layer on a flat plate under the action of acoustic free-stream disturbances at several frequencies simultaneously is also studied. It is demonstrated that combin ation (sum and difference) frequencies appear and interact with each other. Bicoherence diagrams are constructed for the amplitudes of acoustic di sturbances on the basis of pressure fluctuations on the boundary -layer edge. The bispectral analysis confirms the presence of no nlinear interactions. This work was supported by the Russian Foundation for Basic R esearch (Grant No. 09-08-00679), by the Analytical Development Targeted Program (Project No. 2.1.1/3963), and by the Presidium of the Russian Academy of Sciences (Program for Basic Research No. 11/9). REFERENCES
1. 2. Kudryavtsev, A.N., Poplavskaya, T.V., Khotyanovsky, D.V. Application of high-order schemes in modeling unsteady supersonic flows. Mat. Model. 2007, V. 19, No. 7, pp. 39-55. Maslov, A.A., Mironov, S.G., Poplavskaya, T.V., Smorodsky, B.V. Stability of a hypersonic shock layer on a flat plate. Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, 2004, No. 2, pp.16-23.

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.., .., .. , , - (-) . ( - ). . . , . , , , . . . 4000 /. , () . . - .

158


.. .. . .. . 2004 .. : " ". . ( ~ 0.3.) 0.02% . , , , . , , : . ( ) . ( 05 08-18244). ..,. ., .. 0.3 Re < 2000. /1, 2/ , Re* , . Re* ~ 1950 . /3/, :

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1. , Re > ~1000 . Re 1000, . Re < ~1000 . 2. 600 < Re < 3000, . Re = 2870, 1600, 1300 , . 1. .., .., .., .. // . . . 2003. 4. . 47-55. 2. .., .., .. // . . . 2006. 6. . 68-76. 3. .., .., .. , 2000, // . 2009. 5024. 40 c. . . , -- , . , , . -

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. . , . LINEAR STABILITY ANALYSIS OF THE MAIN REGIME OF FLUID MOTION BETWEEN TWO ROTATING PERMEABLE CYLINDERS Romanov M. N. Southern Federal University, Rostov-on-Don It has been investigated the stability of the main regime of motion of viscous incompressible liquid flows between two infinite permeable rotating concentric cylinders in the presence of radial flow directed from the internal cylinder towards the external one. The main regime representing the circular Couette flow with nonzero radial velocity field component can lose stability in two ways. As a result of the superposition of the monoto nous rotationsymmetric disturbances, the Couette flow is replaced by the Taylor -type vortices. The superposition of three-dimensional oscillatory disturbances leads to a three-dimensional flow with azimuthal waves. In the research there have been computed the neutral curves for two types of losing stability of the main r egime. : .. - , ( ) , .. 1 100 , , . - (, , ), .. -

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- . , , . . . , , . , [1­3]; [3­7]; [8­11]; [11]; , [10, 12]; [1 3 ] . « - » ( 230), ( 07-08-00164) -454.2008.1 . 1. 2. 3. 4. 5. .., .. // , 2003, .29. . 13, . 71-79. .., .. // , 2004, . 11, 2, . 247-257. .., .. .. // , 2005, . 10, 6, . 64-96. Rudyak V.Ya., Belkin A.A., Krasnolutskii S.L. Diffusion of nanopart icles in gases and fluids // International J. of Nanomanufacturing, 2008, v. 2, No. 3, p. 204-225. .., .., .. (2008) -

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6. 7.

8. 9. 10. 11. 12. 13.

. - . 2008. . 81, 3. . 76­81. .. . // . . , 2002, . 386, 5, . 624-628. Rudyak V.Ya., Dubtsov S.N., Baklanov A.M. Measurements of the temperature dependent diffusion coefficient of nanoparticles in the range of 295­600K at atmospheric pressure // J. Aerosol Sci., 2009, v. 40, No. 10, p. 833-843. .., .. // . . , 2003, . 392, 4. . 624­ 627. .., .., .. (2009) . . . 79. . 18­25. Rudyak V.Ya. et al. Nanoparticle friction force and effective viscosity of nanofluids // Defect & Diffusion Forum, 2008, v. 273-276, p. 566-571. Rudyak V.Ya. et al. Viscosity and themal co nductivity of nanofluids // Proc. of 2nd Micro & Nano Flows Conf. West London: Brunel University, 2009, p. MNF32-1-MNF32-8. .., .., .. , // , 2008, . 34, . 2, . 69 74. .. . // . . , 2007, . 412, 4, . 490-493. TRANSPORT PROCESSES IN NANOFLUIDS: STATE AND PROBLEMS Rudyak V.Ya. Novosibirsk State University of Architecture and Civil Engineeri ng

Nanofluids are two-phase systems consisting of a carrier medium (gas or liquid) and nanoparticles. Nanoparticles are the particles whose typical size is from 1 to 100 nm, they can generally be solid, liquid, or gaseous. Typical ca rriers are water and organic liquids (ethylene-glycol, oil and other lubricants, bio-fluids), polymer solutions, etc. Typical solid nanoparticles are usually particles of chemically resistant metals or metal oxides. Research on the physics of nanofluids and especially their transport properties has started recently and has been motivate by their various applications in different MEMS - and nanotechnologies. This subject has important fundamental constituent because

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the nanoparticles transport processes in fluids are not describe by the traditional methods in general. In present paper the recent state of studding the nanofluids transport processes is reviewed. The experimental and theoretical data including data obtained by the author and his group are analyzed. In particular, it is consider characteristic mechanisms of nanoparticles transport processes in gases and liquids [1­3]; diffusion of nanoparticles in gases and liquids [3­6]; effective viscosity of gas nano-suspebsions and nano-suspebsions [7­10]; thermal conductivity of nanofluids [10]; the force acting on nanoparticle in fluids [9, 11]; non-classical diffusion of molecules in dense gases and liquids [12]. This work was supported in part by the Russian Foundation for Basic Research (grant No. 07-08-00164), the Program "Scientific and scientificpedagogical personnel of innovative Russia in 2009-2013" of the Ministry of Education and Science of the Russian Federation (project No. P230) and by the grant of the President of the Russian Feder ation for Support of Leading Scientific Schools (project no. NSh-454.2008.1). REFERENCES 1. Rudyak V.Ya., Belkin A.A. Nanoparticle velocity relaxation in co ndensed carrying medium // Tech. Phys. Letters, 2003, v. 29, p. 560-562. 2. Rudyak V.Ya., Belkin A.A. Mechanisms of collective nanopa rticles interaction with condensed solvent // Thermophysics & Aeromechanics, 2004, v. 11, 2, p. 54-63. 3. Rudyak V.Ya., Belkin A.A., Krasnolutskii S.L. Statistical theory of nanoparticle transport processes in gases and liquids // Thermophysics & Ae romechanics, 2005, v. 10, p. 489-507. 4. Rudyak V.Ya., Belkin A.A., Krasnolutskii S.L. Diffusion of nanopa rticles in gases and fluids // International J. of Nanomanufacturing , 2008, v. 2, No. 3, p. 204-225. 5. Rudyak V.Ya. et al. About measurement methods of nanoparticles sizes and diffusion coefficient // Doklady Phys., 2002, v. 47, p. 758 -761. 6. Rudyak V.Ya., Dubtsov S.N., Baklanov A.M. Measurements of the temperature dependent diffusion coefficient of nanoparticles in the range of 295­600K at atmospheric pressure // J. Aerosol Sci., 2009, v. 40, No. 10, p. 833-843. 7. Rudyak V.Ya., Krasnolutskii S.L. About viscosity of rarefied gas suspensions with nanoparticles // Dokl. Phys., 2003, v. 48, p. 583-586.

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8. Rudyak V.Ya., Belkin A.A., Egorov V.V. On the effective viscosity of nanosuspensions // Technical Phys., 2009, v. 54, p. 1102 -1109. 9. Rudyak V.Ya. et al. Nanoparticle friction force and effective viscosity of nanofluids // Defect & Diffusion Forum, 2008, v. 273-276, p. 566-571. 10. Rudyak V.Ya. et al. Viscosity and themal conductivity of nanofluids // Proc. of 2nd Micro & Nano Flows Conf. West London: Brunel University, 2009, p. MNF32-1-MNF32-8. 11. Rudyak V.Ya., Belkin A.A., Tomilina E.A. Force acting on nanoparticle in fluid // Tech. Phys. Letters, 2008, v. 34, p. 76-78. 12. Rudyak V.Ya. et al. On the nonclassical diffusion of molecules of liquid and dense gases // Doklady Phys., 2007, v. 52, p. 115­118. .. . .., ,

R v R 1
. - -

R

, , DNS- LES. , -



0.1 10 ,

. , ,

y p y y (v p ) ,
165




y R

-

. .





.

,

10



0.06R



.

10 0.06R

:

y y R .


10 0.06R -



0.06R

-

- [1] , . [2] [3], , . k-- - [4]. . 1. Zaichik, L.I., Soloviev, S.L., Skibin, A.P. Alipchenkov, V.M. A Diffusion-Inertia Model for Predicting Dispersion of Low-Inertia Particles in Turbulent flow// 5th International Conference on Multiphase Flow, Yokohama, Japan, Paper No. 220, 2004.

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2. .. // . . . 2010. 1. C.84-95. 3. .., .. . ., , 2007. 312 c. 4. Demenkov A.G., Ilyushin B.B., Sikovsky D.Ph., Strizhov V.F., Zaichik L.I. Development of diffusion-inertia model for particle deposition in turbulent flows// Journal of Engineering Thermophysics, 2009, V.18, 1, P.39-48. SIMILARITY LAWS OF NEAR-WALL TURBULENT DISPERSED FLOW WITH PARTICLE DEPOSITION D.Ph.Sikovsky, Institute of Thermophysics SB RAS, Novosibirsk Novosibirsk State University Scaling analysis and method of matched asymptotic expansions is a pplied for the problem of the deposition of small inertial particles from turb ulent dispersed flows at large Reynolds number R v R 1 ( v ­ friction velocity, R ­ outer lengthscale) and the range of the dimensionless part icle relaxation time R . For the diffusion-impaction and inertiamoderated regimes of particle deposition the similarity laws are obtained for the rate of deposition, which are in good agreement with the data of exper imental and DNS and LES data. It is shown that in the diffusion -impaction regime



0.1 10 the effect of Brownian diffusion is non-negligible,

as it was supposed by some researchers. Within the overlap layer, or log layer the particle concentration and statistical moments of particle velocities are shown to be the universal functions of the coordinate y p y y (v p ) , the logarithmic law for the concentration being valid in the range of distances to the wall y R . The composite asymptotic expansions are derived for the particle concentration and the s econd moments of velocity fluctuations which are uniformly valid for .

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It is shown that inertia-moderated regime of deposition 10 are subdivided into two subregimes at large Reynolds number of flow. When 0.06R , then the particle inertia is essential everywhere. In the range

10 0.06R the log layer are divided into two regions: the region of essential effect of the particle inertia y , and the region of negligible particle inertia y R . Such subdivision gives a ground for the

usage of the simplified diffusion-inertia model [1] for the numerical modeling of dispersed flow with relatively inertial particles 10 0.06 R together with the wall functions for the particle concentration being the solution of simplified one-dimensional problem of inertial particle deposition in the near wall region. Such a problem is solved on the base of results from [2] and di fferential particles Reynolds stress transport model [3]. The wall functions for the concentration and the second moments of velocity fluctuations are d erived. The wall functions are approved in numerical simulation of the part icle deposition from the turbulent flow in circular tube u sing the k--model of turbulence and diffusion-inertia model [4]. . 1. Zaichik, L.I., Soloviev, S.L., Skibin, A.P. Alipchenkov, V.M. A Diffusion-Inertia Model for Predicting Dispersion of Low-Inertia Particles in Turbulent flow// 5th International Conference on Multiphase Flow, Yokohama, Japan, Paper No. 220, 2004. 2. .. // . . . 2010. 1. C.84-95. 3. .., .. . ., , 2007. 312 c. 4. Demenkov A.G., Ilyushin B.B., Sikovsky D.Ph., Strizhov V.F., Zaichik L.I. Development of diffusion-inertia model for particle deposition in turb ulent flows// Journal of Engineering Thermophysics, 2009, V.18, 1, P.39-48.

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.. , . .. . ( ) . . . (; ; ). -. i v i f1i t w s t s 4 1 h 1 s w h f 2 , t 3 y y (1) h y h ( y 1) s f 3 t Pr
1
t 0

0,

2 t 0

0,

3 t 0



1 1 g 22 2 2 g12 2 , g
t 0





s

t 0

0,w

t 0

0,h

0.
T T 0 . ,

T w ln , rot , s div ,h T 1, , , , 0 T0 0 , T , , - , , ,

; - ; Pr ­ ; - . f1i , f 2 , f3 - , . , . - l 0 ,
0

c0

- t0 .

0

2 c0

, c . "0" -

169


- . . . . [1]. . 1-3 . Oz . , 3 . . , , , .

. 1.



3

x y t=0.001,

z=0.

. 2.



3

x y t=0.1,

z=0.

170


. 3.



3

x y t=5,

z=0.

1. Morton T.S. The velocity field within a vortex ring with a large elli ptical cross-section// Journal of Fluid Mechanics. 2004. V. 503. P. 247 ­ 271. EVOLUTION OF THE VORTEX RING IN VISCOUS HEAT-CONDUCTING GAS Solenaya O.A. Physics Faculty, M.V.Lomonosov Moscow State University Vortex structures (vortex tubes and rings) play an important role in tu rbulent flows. The motion of vortex ring in ideal fluid was investigated for the first time by Lamb. The limiting case of the vortex ring is the Hill's spherical vortex. Previous investigations were fulfilled under various simplifying a ssumptions (incompressibility; small cross -section of the vortex core; constancy of the viscosity). We used the full system of the Navier-Stokes equations.

i v i f t w s t

1i

171


s 4 1 h s w t 3 y y h h ( y 1) s t Pr
1
t 0

1 h f 2 , y f
3

(1)

0,

2 t 0

0,

3 t 0



1 1 g 22 2 2 g12 2 , g
t 0





s

t 0

0,w

t 0

0,h

0.

T T w ln , rot , s div , h T 1, , , , . 0 T0 0 T0

Here , T , , are density, temperature, viscosity, thermal conductivity, respectively,

is the Laplace operator, Pr is the Prandtl nu mber, is the

adiabatic exponent. The symbols f1i , f 2 , f3 denote nonlinear members (with regard to the first derivatives). The system is a parabolic one, it is written in a dimensionless form. The characteristic length is l0 time is t0
0 2 c0

0

c0

, the characteristic

, being the sound velocity. The subscript "0" refers to the

initial state. We solved the Navier-Stokes equations with the aid of parametrix method. The proposed method of solution of the s ystem consists in reducing it into the system of integral equations of the Volterra type and in subsequent use of iterative procedure. The solution to the linearized system was taken as the first iteration. Calculations have been made referring to the evol ution of the vortex ring. Initial data for the velocity distribution were taken from the paper of T.S. Morton. Figs. 1-3 show the variation of the component of the vorticity. The axis of symmetry of the vortex ring is in the coordinate origin. As seen, th e vorticity component 3 decays as the value of t increases due to dissipation. We calculate components of the velocity of the vortex ring, as well. Thus the problem of evolution of a vortex has been solved numerically. We took into account viscosity, thermal conductivity and compressibility.

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Fig. 1.



3

against x and y at time t=0.001, section of the vortex ring by plane z=0.

Fig. 2.



3

against x and y at time t=0.1, section of the vortex ring by plane z=0.

Fig. 3.



3

against x and y at time t=5, section of the vortex ring by plane z=0.

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REFERENCE 1. Morton T.S. The velocity field within a vortex ring with a large elliptical cross-section// Journal of Fluid Mechanics. 2004. V. 503. P. 247 ­ 271. .., .., .. () - ­ . ( 30% ), , . [1]. , . ( - , , ). , 6. . , . [ 2], . . , , 20% . 7º. 174


[2], 5.95. , 4 , . ( 09 -0800472) 2.1.1/200. . 1. Egorov I.V., Fedorov A.V., Soudakov V.G. Direct numerical simul ation of disturbances generated by periodic suction-blowing in a hypersonic boundary layer // Theoret. Comput. Fluid Dynamics, 2006, 20(1), pp. 41 -54. 2. Fedorov A.V., Kozlov V.F., Shiplyuk A.N., Maslov A.A., Sidorenko A.A., Burov E.V., Malmuth N.D. Stability of hype rsonic boundary layer on porous wall with regular micr ostructure // AIAA Paper 2003-4147, 2003. - .., .. , ; , . , , , . (E.I.Suetnova, Guy Vasseur, 2000). , , .

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, , . . E.I.Suetnova, Guy Vasseur. 1-D Modelling rock compaction in sedimentary basin using visco-elastic rheology // Earth and Planet. Sci. Letters, 2000, V. 178. P. 373-383. : . .., .. . , , , [1]. -. . , ­ [2]. , . [2]: . [4] , .

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. , , . . ( [5]) , «». 08 -08-00463. 1. . . . , . 425, . 3, . 334­337, 2009. 2. .. . . 2008. . 418. 1. . 42-45. 3. .. . . , . 5, . 110, 2004. 4. .., .. . . , . 3, . 114-119, 2009. 5. . : .. 29- 18 2010 .., .. , - , «»- .

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( - ~10-3), - - . [1] <<1 , , ( ), -, ­ . , , ­ - , . , . 08 -0800463. 1. Yuri M. Shtemler, Michael Mond, Vladimir Cherniavskii, Ephim Golbraikh, nd Yaakov Nissime An asymptotic model for the Kelvin­Helmholtz and Miles mechanisms f water wave generation by wind. PHYSICS OF FLUIDS 20, 1, 2008. .. , , , , , . - [1] . , . (; .. ) ( ) 2000,

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( ). 5 : , ; , ; "" . () ( ), ( ); , , . . () ( ) 21 (13 8 ) , . , , , , 1/3 , . . ­ [2]. , (SNIC), . , , (9 ) ( ). - ( , , ) [3]. .. , .. . [4]. "FundaÃÖo para a CiÉncia e a Tecnologia" (), SFRH/BD/23161/2005.

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1. .. -: // , 1999. 2. Golubitsky M., Schaeffer D.G., Stewart I. Singularities and groups in bifurcation theory, Volume II // Springer, 1988. 3. Zheligovsky V. Amplitude equations for weakly nonlinear two -scale perturbations of free hydromagnetic convective regimes in a rotating layer// Geophysical & Astrophysical Fluid Dynamics, 2009, v. 103, iss. 5, p. 397 ­ 420. 4. Chertovskih R., Gama S., Podvigina O. and Zheligovsky V. Dependence of magnetic field generation by thermal convection on the rot ation rate // Physica D [http://arxiv.org/abs/0908.1891]. CONVECTIVE DYNAMO IN ROTATING LAYER Chertovskih R.A. University of Porto, Faculty of Sciences, Department of Mathematics, Porto, Portugal International Institute of Earthquake Prediction Theory and Mat hematical Geophysics, Moscow In this work the influence of the rate of rotation on magnetic field gene ration by convective flows in a rotating plane layer is explored numerically. Rayleigh-BÈnard convection [1] is considered in the Boussinesq approximation, the horizontal boundaries are stress-free, isothermal and perfectly electrically conducting. We have determined types of the hydrodynamic ( HD) convective attractors (in the absence of magnetic field) in a plane layer of rotating fluid with square periodicity cells for the Taylor number varied from zero (no rotation) to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We have observed 5 types of the HD attractors: two families of rolls of different widths, dep ending on two spatial variables and aligned with a side of periodicity boxes; rolls, parallel to the diagonal; travelling waves; and fully three-dimensional "wavy" rolls. For this flows the kinematic (linear) dynamo was solved for magnetic Prandtl number slightly larger its critical value (at which magnetic field ge neration sets in); all types of the HD attractors, except for a family of rolls, are found to be capable of kinematic magnetic field generation. In dominant magnetic modes the field concentrates near horizontal boundaries in flattened halfropes. For the same parameter values in nonlinear regime (magnetic field a ffects fluid flow by the Lorentz force), we have found 21 distinct convective

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magnetohydrodynamic (MHD) attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge and disappear. The nonlinear MHD convective system demonstrates rich dynamics when the Taylor number is varied: we have also observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an inco mplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The nonlinear MHD system is highly symmetric. Bifurcations in the presence of symmetries are the subject of the equivariant bifurcation theory [2]. We have found two novel global bifurcations reminiscent of the sa ddle-node on invariant circle (SNIC) bifurcation, which are only possible in the presence of symmetries. We have also found nonlinear convective MHD regimes po ssessing the symmetry about a vertical axis (9 steady states and a time -periodic regime) and parity-invariant regimes (a stead y and a time-periodic one). These symmetries of perturbed states are essential to guarantee insignificance of the -effect in the leading order in the evolution of their li near and weakly non-linear perturbations (otherwise generically large -scale perturbations grow superexponentially) [3]. Our results do not challenge the universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid at even higher rates of rotation), but we demonstrate that this picture lacks many significant details. This work was done in collaboration with V.A. Zheligovsky, O.M. Podvigina and S. Gama, its results are presented in details in [4]. The author was financially supported by the "FundaÃÖo para a CiÉncia e a Tecnologia" (Portugal), grant SFRH/BD/23161/2005. REFERENCES 5. Getling A.V. Rayleigh-BÈnard convection: structures and dynamics // World Scientific, 1998. 6. Golubitsky M., Schaeffer D.G., Stewart I. Singularities and groups in bifurcation theory, Volume II // Springer, 1988. 7. Zheligovsky V. Amplitude equations for weakly nonlinear two -scale perturbations of free hydromagnetic convective regimes in a rotating layer// Geophysical & Astrophysical Fluid Dynamics, 2009, v. 103, iss. 5, p. 397 ­ 420. 8. Chertovskih R., Gama S., Podvigina O. and Zheligovsky V. Dependence of magnetic field generation by thermal convection on the rot ation rate // submitted to Physica D [http://arxiv.org/abs/0908.1891].

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..,. . , [1], Re = 300­500 , . -, , , , ­ [1]. « ». - ­ , . [2], . . . , : ) ; ) -; ) -, , . , . , [1]. 1. hang H.-C. and Demekhin E.A. Complex wave dynamics on thin films / Elsevier, 2002, 400 p. 2. .. / .: , 1990, 230 .

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. ., .., .., .. . .. . . , . 2

4060 . , 700 . = 2-4. . ( 10 65 ) . -, [1,2]. . , [1,2] . , , . .. , . . ( -2.5 ) . 2% . . 1 , . ( ) , ( ) ­ . , 2 M 3.4 , . . . , . , .

183


. , , . 1. .. , .. . « ». ., «», 1977. 2. « ». ., «», 1964.

P

3 .0

2 .5

2 .5

3 .0

3 .5

4 .0

M
. 1.

VALUE OF PRESSURE BEHIND THE SHOCK PROPAGATING THROUGH A TURBULENT AIR FLOW Shtemenko L.S., Dokukina O.I., Shugaev F.V., Terentiev E. N. Physics Faculty of M.V. Lomonosov Moscow State University The pressure was measured behind a plane shock propagating through a turbulent air flow. The measured values are larger as compared with the data referring to the laminar flow behind the shock wave at the appropriate cond itions. The mechanism is set forth which accounts for this phenomenon. The experiments were fulfilled in a shock tube of a rectangular 40x60 mm cross184
2


section. We used grid-generated turbulence. The grid was situated at a distance of 700 mm from the test section. The Mach numbers of the incident shocks were M=2-4. The incident shock reflects from a perforated plate at the end of the test section and hits the turbulent flow. We measured the mean v elocity of the incident shock as well as of the reflected one at a base whose size was equal 10--65 mm at some distance from the end of the shock tube. A configuration appears while the reflected shock interacts with the boundary layer, and the shock becomes concave. The data concernin g the velocity of the reflected shock are somewhat contradictory. The calculations by Mark show the decrease of the velocity meanwhile other authors state that it increases [1,2]. Our data show that at first the velocity grows and then decreases. In other words, there is a shock focusing at first and then the shock becomes divergent. Further on the process may be repeated. A pressure transducer of 2.5 mm diameter was installed into the wall of the shock tube at a distance of three bores from the end of the tube. The amplitude of turbulent fluctuations was 2% of the pressure behind the incident shock. Fig.1 shows the dimensio nless pressure (the ratio of the value behind the reflected shock and the appr opriate value ahead of the shock) against the Mach number of the incident shock. Upper graph (small circles) corresponds to the turbulent flow, lower graph (large circles) corresponds to the case when there is no turbulence. As seen, the pressure is greater within the interval of Mach numbers 2 M 3.4 in the first case as compared with the second one. The possible mechanism is as follows. There are acoustic waves in the turbulent flow behind the shock wave. Turbulent structures become unstable after the passage of the shock, and new acoustic waves appear. The acoustic waves heat the gas, and the pressure rises as a result. The time of the passage of the shock arises, while the Mach number increasing. The heat flux diminishes and then disappears. REFERENCES 1. T. V. Bazhenova, L.G. Gvozdeva. Nonstationary interactions of the shock waves. M., " Nauka", 1977 (in Russian). 2. "Physical gasdynamics and gas properties at high temperatures". M., "Nauka",1964.

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P

3 .0

2 .5

2 .5

3 .0

3 .5

4 .0

M
Fig.1. , .. , - , , . , , . , , , . , , . -

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2003-2008 . Hydrodyna (Eureka, no. 3246) , () : GE Hydro Division (), Vatech Hydro A.G. (), Voith Siemens Hydro Power Gener ation Gmbh & Co (). , Hydrodyna , [1-3] Re=Uh/ , * H=*/** ( U - , h - , - , **- ). , , , , . , . SINF [4], 15 . , k- k- . , [2]. (5103 1 - H «» , 1.8, ( ). , , - H . , ( */h ~ 0.1), - . 187


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