7 TH EUROPEAN CONFERENCE FOR AERONAUTICS AND SPACE SCIENCES (EUCASS) Dusty Plasma Voi Dynamics in Unmoving an Moving Flows O.V. Kravchenko*, O.A. Azarova**, an T.A. Lapushkina*** *Scientific an Technological Center of Unique Instrumentation of RAS Russia, Moscow, 734, Butlerova str. 5 **Doronicyn Computing Centre, Feeral Research Center Computer Science an Control of RAS Russia, Moscow, 9333, Vavilova str. 4 ***Ioffe Institute Russia, Saint Petersburg, 94, Politekhnicheskaya str. 6 Abstract Generation of steay structures with empty regions (vois) has been moelle numerically in unmoving an moving flows of usty plasma. The simulations are base on the known moel of Avinash, Bhattacharjee an Hu of formation of a voi in a usty plasma flow. The moel has been reuce to the ivergent form an two algorithms of first an secon orers for calculations have been suggeste. The La s scheme an the comple conservative ifference scheme have been use for the first an secon orer for the hyroynamic part of the moel. Results on the ynamics of vois have been obtaine at the stage of circular ring structure generation an at the final stage of a steay roun voi formation. Behaviour of the efining flow parameters has been obtaine up to the steay usty plasma flow moe. Description of the eperimental installations for investigations of usty plasma moving an unmoving flows an generation of structures in these flows has been presente.. Nomenclature n, n e = concentrations of ust an electrons components v, v i = velocities of ust an ions components E = electric fiel currency F = ion-rag force D = iffusion coefficient t,, y = time an spatial variables a, b = fit parameters of ion-rag force approimation µ = coefficient of ions mobility α = friction coefficient τ, τ i = coefficients of normalize temperature for ust an ion species = amount of the gri points in the -irection n i Ine refers to the initial parameters DC Direct current. Introuction A number of the investigations of the ynamics of usty plasma were publishe in recent years [, ]. At the same time, it is not so much stuies in which the numerical simulations of the structure generation in usty plasma are performe. The first usty plasma structure containing empty omains (vois) was iscovere in the course of the eperiments on the boar of the International Space Station []. Also, in the laboratory conitions the vois have been foun later [3-5]. Evolution of the ynamics of a single symmetric voi from equilibrium was escribe by electro-hyroynamic moel [6, 7] taking into account the effect of an ion attraction force as a nonlinear function of the spee of ions. The algorithm for simulations of appearance of Copyright 7 by O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina. Publishe by the EUCASS association with permission.
O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina usty plasma voi using the moel [7] has been presente in [8] for the case of cylinrical geometry of the electrical fiel. Generation of a single symmetric voi an concentric symmetrical vois in unmoving flows of low-temperature usty plasma has been obtaine in [9]. A stuy of generation of vois in moving flows together with the further research of the vois in unmoving meia is a purpose of this paper. The simulations are base on the moel [7] of formation of a voi in the usty plasma flow. The La s scheme an the comple conservative ifference scheme [] have been use in the first an secon orer approaches for the hyroynamic part of the moel. 3. Methoology 3. Description of the using physical moel Simulations are base on the ifference schemes of the first an secon orer of approimation of the moel [7]. Here the moel is consiere in imensionless form, believing that the normalizing parameters from [7] have been applie: v v n t n v F E v, () n ( n v ) n = D t, () n E F n E e e, (3) i n e n, (4) ae, b v v E 3 i. (5) i Here n, n e are concentrations of ust an electrons components, v, v i are velocities of ust an ions components, E is electric fiel currency, F is ion-rag force, D is iffusion coefficient. In fact the governing equations in this moel contain ust momentum () an ust continuity equations (), balance equation on electrons neglecting the electron inertia (3). Poisson law which completes the nonlinear system of equations, an the epression for ion rag force are presente in (4, 5). It shoul be notice that ion rag force (5) is a force which acts on ust particles from ions an is approimate by nonlinear function with fit positive constants a, b. Equations () an () constitute the hyroynamic part of the moel an the equations (3)-(5) are the electrostatic part of it. One of the major features of the moel [7] is that ion rag force F epens on the ion velocity v i an it ecreases with the increase of the velocity as a cubic function of v i. Also, this moel is a self consistent system of partial ifferential equations in which the ynamics of the charge component of meium effects on electrical fiel istribution E an vise versa electrical fiel istribution effects on a motion of charge particles. One coul realize that ensities of ust particles, electrons an ions are the unknown functions as well as electrical fiel istribution function E. Velocity of ions v i is linearly proportional to electrical fiel istribution E with coefficient of proportionality µ. 3. Description of the numerical moels with the first an secon orer of ifference schemes for hyroynamic part of the moel A general algorithm of the numerical simulation for the first orer moel from [7] () (4) which was introuce recently [8] is as follows:
DUSTY PLASMA VOID DYNAMICS Computation of the initial value problem of (), () with Neumann bounary conition by eplicit conservative scheme (the La s scheme in the first orer case) on current time step of integration t n, so n, v are known functions on the current time step. Computation of the initial value problem of (3),(4) by eplicit solver of Cauchy problem on the same time step of integration t n+, so n e, E are known functions on the current time step t n also. Here the metho of Runge-Kutta of the forth approimation egree is use. Recomputation of (), () with Neumann bounary conition on the net time step of integration t n+ with respect to previous items, so n, v are known functions on the net time step t n+. For the first orer calculations of the hyroynamic part of the moel the La s scheme is use (with D =) for the ivergent form of the equations (), (): u F t f, (6) where n u v, F.5v n v ln n, f F n D E The secon orer scheme for the algorithm [8, 9] was obtaine (without recomputation) using the approach of [] for increasing the ifference scheme orer. By this way the systems of ivergent equations (6) are use for unknown functions together with the system for their space erivatives: where u n v, F u t F (.5v, ( n ln n D n ) ) E F In the systems of the erivatives (7) the space erivatives from the right parts are inclue into the flu functions. It provies the absence of the numerical (parasite) sources an runoffs. Equations (3), (4) are approimate with the secon approimation orer using the central form for the space erivatives an the metho of Runge-Kutta of the fourth orer of approimation is use, too. In the numerical simulations the net restrictions were impose: if n > then n was applie by an v was applie by v. 4. Numerical simulations Numerical eperiments on generation of vois in the flows of usty particles have been conucte for zero an nonzero initial velocity v of a usty component. Parameters of the simulations are collecte in Tab.. 4. Comparison of the simulations using the first an the secon orer moels Comparison of the simulations using the ifference schemes of the first an secon approimation orer is presente in Fig.. It can be seen that the obtaine ynamics are quite the same (Fig. a). It shoul be note that the restrictions on v is introuce for cutting the high frequency numerical oscillations in the ynamics of v (when n becomes more than ). These oscillations are seen in the profile of n calculate with the use of the secon orer scheme (Fig. b, re curve). Introuction into the scheme the term with the iffusion of concentration of the usty component (which plays the role of viscosity in ()) allowe smoothing these oscillations (Fig. b, black curve, D =.). Note that the iffusion term is introuce into the ifference approimations via the flu function in (7). v v v.. (7) 3
O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina.5 n n.8.5.6.4.5. 4 6 8 4 6 8 a) b) Figure : Comparison of ynamics of concentration of usty component n uring the voi formation calculate with the use of first (blue curve, D =) an secon (re curve, D =) orer schemes for the hyroynamic part of the moel, n =.3, v =, : a) beginning stage (without the restrictions on n an v ); b) subsequent stage (with the restrictions on n an v ), black curve - D =. a) b) c) ) Figure : Dynamics of concentration of usty component n uring the voi formation, E = 4-4 (first orer scheme): a) - non-imensional time t=7 (unsteay ring structure); b) - t=; с) - t=4; ) - t= (steay voi) 4
DUSTY PLASMA VOID DYNAMICS 4. Moeling of ynamics of generation of voi in unmoving flows Dynamics of generation of a voi in the unmoving flow (v =) of usty particles with the use of the first orer algorithm for the hyroynamic part is shown in Fig.. Two-imensional figures for n obtaine by the rotation technique are presente. The mechanism of generation of vois is connecte with the superposition of the electrical fiel an the ion attraction force action. Voi grows in time from the constant parameters an becomes saturate at time moment t= (Fig. ). The transitional states are presente in Figs. a-c at time moments t=7,, 4, respectively. It can be seen that at the first stage the ring structure is forming with a voi generate in a centre of it. Then the right bounary of this structure moves right an leaves the calculation area keeping the constant parameters behin it. a) b) c) Figure 3: Depenence of n istributions on initial value of the electric fiel E, t= (first orer scheme): a) E = 4-6 ; b) E = 4-5 ; c) E = 4-4 5
O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina ) Figure 3 (continuation): Depenence of n istributions on initial value of the electric fiel E, t= (first orer scheme): ) E = 4-3 Distribution of n for ifferent initial istributions of the electric fiel E is presente in Fig. 3. Here the beginning stage with generation of the ring structure of usty particles is presente. One coul fin that the size of a voi region ecreases while the initial value E increases from -6 (Fig. 3a) to -3 (Fig. 3). For E = -6 the voi s raius is close to 4 (Fig. 3a) while it is about. for E = -3 (Fig. 3). From the other han, it coul be foun that there is a semi-saturate state of the voi s formation insie the generate ring structure of usty particles. Table : Simulation parameters of the numerical eperiment Parameters Values τ i.5 τ. a 7.5 b.6 α µ.5 n e.999 Dynamics of generation of a voi with the use of the secon orer ifference scheme for the hyroynamic part of the moel [7] is shown in Fig. 4 (here D =., b=). The blue curve presents the profile of n at thebeginning stage of the unsteay concentric ring structure formation an the green an re curves correspon to the stage of the steay voi formation in the central area (Fig. 4a). Figure 4b emonstrates the profiles of all the efining parameters for the steay flow containing the forme voi. Two-imensional figures for n obtaine by the rotation technique are presente in Figs. 4c, 4. It can be seen that the presence of iffusion of the concentration increases the values of n at the voi s center (Fig. 4c) but by changing the the constant b in the epression of the ion-rag force F (5) the value of resiual usty component concentration can be ecrease into the voi (Fig. 4, b=.4), the ecrease of n being obtaine to provie some icrease of the voi s raius. 4.3 Moeling of ynamics of generation of voi in moving flows Dynamics of generation of a voi in the moving flow (v =.) of usty particles with the use of the first orer scheme for the hyroynamic part of the moel [7] is presente in Fig. 5. Here all the efining parameters are shown at the stage of establishing the steay flow moe (Figs. 5a, 5b). The voi which was forme at this quasi-steay moe is shown in Fig. 5c using the rotation technique. Dynamics of the voi bounary (curve ) an the value of v for (n i) (curve ) are presente in Fig. 6. It can be seen that the velocity of the usty 6
DUSTY PLASMA VOID DYNAMICS n t=.8.6 F E.6. n.8 n e.4..4 v 4 6 8 a) b) -.4 4 6 8 с) ) Figure 4: Simulations with the use of secon orer scheme, n =.3, b= (a) (c), n =., b=.4 (), v =, E = 4-5, n i =, D =.: a) - ynamics of concentration of usty component n uring the a voi formation: t= (blue), t=6 (green), t= (re); b) - profiles of the efining flow parameters on the stage when the steay voi is forme, t=; c), ) - steay vois obtaine (n ), t= component (curve ) performs the oscillations of small amplitue which is connecte with the applie restriction on v. Nevertheless the voi bounary (curve ) is shown to be a converging curve. It has to be note that it coul be interesting to provie such type of the restrictions which woul cause the oscillations of the voi s bounary. Dynamics of generation of a voi in the case of v =.3 is presente in Fig. 7. Here all the efining parameters at the stage of establishing the steay flow moe are shown (Figs. 7a) an the voi obtaine at this quasisteay moe is shown in Fig. 7b using the rotation technique. It is seen the non-monotonic behavior in the istribution of concentration of usty particles n which may be connecte with a simple waves effect in periphery area of the voi. 7
O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina.5 t=4 E.5 t=4 E F F.5.5 n e n n e n.5.5 v -.5 4 6 8 -.5 a) b) v 4 6 8 c) Figure 5: Profiles of the efining flow parameters on the stage establishinh the steay voi formation, first orer scheme, n =., v =., E = 4-5, n i =, b=: a) - t=4 ; b) - t=4; c) istribution of n in quasi-steay flow moe, t=4 8
DUSTY PLASMA VOID DYNAMICS 6 v, v (n i ) 4-3 4 t Figure 6: Dynamics of the voi s bounary (curve ) an the value of v in (n i ) (curve ), first orer scheme, n =., v =., E = 4-5, n i = 3 t=4 E F n e n v 4 6 8 a) b) Figure 7: a) - profiles of the efining flow parameters at the stage of the steay voi formation, first orer scheme, n =., v =.3, E = 4-5, n i =, t=4; c) quasi-steay voi obtaine (n ), t=4 9
O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina 5. Eperimental installations (principal schemes an escription) Here two eperimental installations are escribe for planne investigations of ynamics of usty plasma flows an formation of usty plasma structures incluing generation of vois in unmoving an moving flows. 5. Eperimental installation for investigation of usty plasma flows in unmoving meia For investigation of usty plasma incluing generation of vois in unmoving meia the net eperimental set up is planne to be evelope on a base of the eisting one. The installation for investigation of plasma ynamic in unmoving meia (Fig. 8) consists from the working chamber with iameter 3 mm an height 4 mm. Two conical electroes are ispose vertically along the camber ais at the istance mm from each other. DC voltage which applie to the electroes prouces a stationary glow gas ischarge at the varying pressure in changeable gases incluing a gas with ust particles. It allows proucing usty plasma in unmoving meia. In the ischarge region the gas temperature is about 5 K, electron temperature is about K an ionization egree is about -6. The electro-ischarge shock tube with iameter 4 mm an length 7 mm is ispose at the lateral wall of the chamber. It allows proucing shock wave with velocity about.3 km/s, at Mach number 3.8. The installation has the electrical probe, piezo probe for pressure measurement, the optical Schliren system an evices for optical an spectrographic measurements. Figure 8: The installation for investigation of usty plasma ynamics in unmoving meia (photo) 5. Eperimental installation for investigation of usty plasma moving flows For investigation of ynamics of usty plasma incluing generation of vois in moving flows the net eperimental set up is planne to be evelope on a base of the eisting one. The shock tube is collecte from the stainless steel wele sections of the rectangular cross section 55 mm, m length (Fig. 9). The plates are polishe an the sections are wele, which allows getting the flow with the small perturbations of the gasynamic parameters behin a shock wave front. The general length of shock tube is m. As the shock tube consists of some sections the relative sizes of the high pressure chamber an low pressure channel can be change epening on require work regime. For creation of shock waves with Mach numbers M > 7, the gas in the high pressure chamber can be aitionally heate up to 4 C. The rectangular cross section of the tube allows stuying the stationary homogeneous non-ivergent flows of the ifferent gas composition in the wie iapason of Mach numbers incluing the flows with aitional injection of micro particles. So it is possible to create the homogeneous ionize gas ischarge zones in the flow
DUSTY PLASMA VOID DYNAMICS with micro particles an to form a flow of usty plasma. In future it will allow investigating the appearance an changing new gas ynamic structures an generation of vois in moving usty plasma flows. Figure 9: The installation on the base of the rectangular cross section shock tube for stuy of moving usty flows (photo) 6. Summary Known moel of Avinash, Bhattacharjee, an Hu of formation of a voi in the usty plasma flow has been rewritten in the ivergent form an the numerical algorithms of the first an secon approimation orer for the hyroynamic part of the moel have been evelope. Numerical simulations with the use of the obtaine numerical moels were conucte an the results on the voi generation an ynamics have been obtaine in unmoving an moving flows of usty plasma. Depenence of the concentration of usty particles on the initial value of the electric fiel was obtaine. It has been shown that the size of a voi region ecreases when the initial value of the electric fiel increases. Results on behavior of the profiles of all the efining parameters at the stage of the concentric ring structure formation an at the stage of the forme voi eistence in the steay flow moe have been obtaine. Description of the eperimental installations for future eperimental investigations of usty plasma moving an unmoving flows an generation of structures in these flows (incluing formation of the vois) has been presente. References [] Molotkov, V., H. Thomas, A. Lipaev, V. Naumkin, A. Ivlev, an S. Khrapak, 5. Comple (usty) plasma research uner microgravity conitions: PK-3 Plus Laboratory on the International Space Station. International Journal of Microgravity Science an Application. 35(3): 8. [] Fortov, V.E., G.E. Morgill. 9. Comple an usty plasmas: from laboratory to Space. CRC Press: 4. [3] Feng, H., Y. Mao-Fu, W. Long, an J. Nan. 4. Vois in eperimental usty plasma. Chin. Phys. Lett. (): 4.
O.V. Kravchenko, O.A. Azarova, an T.A. Lapushkina [4] Sarkar, S., M. Monal, M. Bose, an S. Mukherjee. 5. Observation of eternal control an formation of a voi in cogenerate usty plasma. Plasma Sources Sci. Technol. 4(3): 7. [5] Feng, H., Y. Maofu, an W. Long. 6. Pattern phenomena in an RF ischarge usty plasma system. Science in China Series G: Physics, Mechanics & Astronomy. 49(5):588 596. [6] Ng, C.S., A. Bhattacharjee, S. Hu. Z.W. Ma, an K. Avinash. 7. Generalizations of a nonlinear flui moel for voi formation in usty plasmas. Plasma Phys. Control. Fusion. 49:583 597. [7] Avinash, K., A. Bhattacharjee, an S. Hu. 3. Nonlinear theory of voi formation in colloial plasmas. Phys. Rev. Lett. 9(7): 4. [8] Kravchenko, O.V., an V.I. Pustovoit. 6. Numerical simulation of ynamics of concentric usty plasma structures. In: 5 th International Workshop on Magneto Plasma Aeroynamics. 8. [9] Kravchenko, O.V. Simulation of spatially localize usty plasma structures in comple plasma. 6. Nanostructures. Mathematical physics an moeling. 5():5 6 (in Russian). [] Azarova, O.A. Comple conservative ifference schemes for computing supersonic flows past simple aeroynamic forms. 5. J. Comp. Math. Math. Phys. 55():67 9.