Numerical and experimental study of the time dependent states and the slow dynamics in a von Kármán swirling flow

Transcription

1 Numerical and experimental study of the time dependent states and the slow dynamics in a von Kármán swirling flow E. Crespo del Arco, E. Serre, J. J. Sánchez-Álvarez, A. de la Torre, and J. Burguete UNED, Madrid Spain UMR 6181 IMT Château-Gombert, Marseille, France E. T. S. I. Aeronáuticos, U. P. M., Madrid, Spain Universidad de Navarra, Pamplona, Spain 1

2 System and basic flow ΩR Re = ν H Γ=. R 2, 2 Vortex dynamics from quantum to geophysical scales. EUROMECH Colloquium 491

3 Previous (recent) results: Analysis of stability and numerical results Re<500 Nore, Tuckerman, Daube and Xin JFM 2003, Experimental results Re<500, Nore, Moisy and Quartier, Phys. Fluids, 2005 Numerical results Re<700, Lackey and Sotiropoulos Phys. Fluids 2006 Experimental results, 10 3 <Re<10 5, de la Torre and Burguete PRL 2007 Motivation: Study of the hydrodynamics in the dynamo action 3

4 Numerical results: Steady flows up to Re=1000 Time dependent up to Re=3000 Experimental results 10 3 <Re<10 5 Effect of forcing the system 4

5 MATHEMATICAL AND NUMERICAL MODELLING u 1 u 2 i u p u + = + t Re iu = 0 Time, velocity units are [Ω 1,ΩR], +Ω u = 0 in r = 1 u = re in z = 0,1 θ ΩR Re = ν H Γ=. R 2, -Ω 5

6 MATHEMATICAL AND NUMERICAL MODELLING + = p + t Re iu = 0 u 1 u u 2 i u, u = 0 in r = 1 u= re in z = 0,1 θ Time discretization: semi-implicit second-order scheme - Backward Euler scheme for the diffusive terms - Adams-Bashforth extrapolation for the non-linear convective terms 3V 4V + V 2 δt. V n+ 1 = 0 in χ Vn+ 1= Wn+ 1 on γ n+ 1 n n 1 n n V. V V. V = pn+ 1+ Δ Vn+ 1+ Fn+ 1 in ( ) ( ) ν χ (65x48x65) en (r,θ,z) The velocity-pressure coupling is treated with a projection scheme - An improved version of the Goda s scheme (Serre & Pulicani, 1998) Spatial approximation: solution is expanded in Chebychev's polynomials in the directions (r, z) and Fourier in the azimuthal direction, θ. 6

7 Experimental diagram of regimes m=2, steady Time dependent m=1, steady Re C =349 M Aspect ratio Nore et al., Phys. Fluids,

8 Symmetry properties of the steady states The basic state is invariant under these operations: R φ : (r, θ,z) (r, θ + φ, z), (u,v,w) (u,v,w) σ: (r, θ,z) (r, θ, 1 - z), (u,v,w) (u,v,-w) - The state after it loses stability, have the smaller symmetry Zπ. - The subsequent instability has the symmetry Z π/2. 8

9 Characteristic time scales Rotation rate Ekman time Viscous diffusing time * t =Ω, t = Ω 1, * R 1/2 tek =, t = Re, Ων R ν 2 * tν =, t = Re 9

10 The flow of pure Fourier modes m=0, m=1, m=2 U U Re=300, m=0 10 Re=390, m=1 Re=440, m=2 Vortex dynamics from quantum to geophysical scales. EUROMECH Colloquium 491 W W

11 Time behaviour of the amplitudes at Re=355 and 440 1,20E-03 2,50E-03 m=2 1,00E-03 8,00E-04 m=1 m=1 2,00E-03 1,50E-03 m=1 m=2 m=3 m=4 6,00E-04 m=2 m=3 m=4 1,00E-03 4,00E-04 2,00E-04 m=2 m=1 5,00E-04 m=2 0,00E+00 0,00E+00 1,00E+01 2,00E+01 3,00E+01 4,00E+01 5,00E+01 6,00E+01 7,00E+01 0,00E+00 0,00E+00 5,00E+01 1,00E+02 1,50E+02 2,00E+02 2,50E+02 3,00E+02 3,50E+02 Re=355 Re=440 11

12 Flow at Re=1000 m=0+1+2 Kinetic energy U W w 12

13 Steady states: azimuthal modes and cat s eyes Re=390, r=0.75, velocity field in the (θ,z) plane One cat s eye Azimuthal modes m=0 and m=1 13

14 Re=1000, two cat s eyes Re=1000, r=0.75, and r=0.9 velocity field in the (θ,z) plane Azimuthal modes m=2 14

15 Cat s eye vortical structure The 3D vortical structure is very complex In the figure Γ=1, Re=500 Lackey and Sotiropoulos Phys. Fluids

16 Re=

17 Amplitude of the modes at Re=1500 2,50E-02 m=0 2,00E-02 1,50E-02 1,00E-02 m=2 5,00E-03 0,00E+00 0,00E+00 2,00E+01 4,00E+01 6,00E+01 8,00E+01 1,00E+02 1,20E+02 m=4, 6 T*=4.8 T w 17 Vortex dynamics from quantum to geophysical scales. EUROMECH Colloquium 491

18 Re=

19 2D Velocity field in two meridian planes 19

20 Instantaneous velocity field Vertical velocity Radial velocity 20

21 Hydrodynamic of a von Kármán flow Flow inside a cylinder with counter-rotating container with propellers. (a) H (b) Dimensions: D y z (c) Freq. Propellers: D y x Fluid: Water at 21º C South North 21

22 Hydrodynamic of a von Kármán flow Re = There are two possible states: vortex travelling up or down 22

23 von Kármán mean flow 100 r (mm) Mean velocity field obtained by LDV 5 Re = z (mm) r (mm) Determination of the state: z (mm) 23 Vortex dynamics from quantum to geophysical scales. EUROMECH Colloquium 491

24 von Kármán instantaneous flow Instantaneous velocity obtained by LDV PDF of the instantaneous velocity 5 Re = The time of escape is 0.2 tν (a) v θ (m/s) (b) de la Torre & Burguete PRL Probability density function (log10) S N N Acquisition time (x10 s) Azimuthal velocity u =v /V θ θ Re S propeller 24

25 Vortex velocity T= 4.2 T W 25

26 When the flow is symmetric with respect to the equatorial plane: 26

27 When the flow is not symmetric with respect to the equatorial plane: 27

28 Forcing the system Static forcing: Ω+Δ 0 Ω 28

29 Forcing the system Periodic forcing: (b) v θ (m/s) Acquisition time (x10 s) 2 29

30 Re=1500 A =A A 0 cos(ωt/100) Axial velocity in the equatorial plane time Alternations between two states are not observed The vortices do not travel t Ek = 38.7 t ν =

31 Conclusions The basic flow loses stability, to a state with a smaller symmetry, and the subsequent transition is to a symmetry state Z π /2 The first time-dependent state is an oscillatory cat s eye pattern which stands at the same position. The vortices travel in experiments and do not travel in numerical simulations, the explanation is in the mean flow <v> θ which is symmetric or not with respect to the equatorial plane. The time for the alternation between the two states (N or S) is about 0.2 tν. The frequency of the vortex is about f w /4.2 in experiments and about f w /6 in numerical simulations indicating that the phenomena might be the same. Z π 31