Nonlinear Convection of Electrically Conducting Fluid in a Rotating Magnetic System H.P. Rani Department of Mathematics, National Institute of Technology, Warangal. India. Y. Rameshwar Department of Mathematics, Osmania University, Hyderabad. India. J. Brestensky and Enrico Filippi Faculty of Mathematics, Physics, Informatics, Comenius University, Slovakia.
Session: GD3.1 Core Dynamics Earth's core structure, dynamics and evolution: observations, models, experiments. Online EGU General Assembly May 2-8 2020
1 Abstract
• Nonlinear analysis in a rotating Rayleigh-Bernard system of electrical conducting fluid is studied numerically in the presence of externally applied horizontal magnetic field with rigid-rigid boundary conditions. • This research model is also studied for stress free boundary conditions in the absence of Lorentz and Coriolis forces. • This DNS approach is carried near the onset of convection to study the flow behaviour in the limiting case of Prandtl number. • The fluid flow is visualized in terms of streamlines, limiting streamlines and isotherms. The dependence of Nusselt number on the Rayleigh number, Ekman number, Elasser number is examined.
2 Outline
• Introduction • Physical model • Governing equations • Methodology • Validation – RBC – 2D – RBC – 3D • Results – RBC – RBC with magnetic field (MC) – Plane layer dynamo (RMC)
3 Introduction • Nonlinear interaction between convection and magnetic fields (Magnetoconvection) may explain certain prominent features on the solar surface. • Yet we are far from a real understanding of the dynamical coupling between convection and magnetic fields in stars and magnetically confined high-temperature plasmas etc. Therefore it is of great importance to understand how energy transport and convection are affected by an imposed magnetic field: i.e., how the Lorentz force affects convection patterns in sunspots and magnetically confined, high-temperature plasmas. • Magnetoconvection exhibits a rich variety of behavior when the
magnetic Prandtl number (Pm) is small. This condition is satisfied in the solar convection zone near the sunspot and magnetically confined, high temperature plasmas. • Convection in planetary cores and stellar interiors often occurs in the presence of strong rotational and magnetic constraints. 4 Motivation • To link the dynamics of geophysical fluid flows with the structure of these fluid flows in physical space and the transitions of this structure arising in the system due to the following parameters: ✓ Rayleigh Number, Ra ✓ Nusselt Number, Nu ✓ Chandrasekhar Number, Q ✓ Taylor Number, Ta ✓ Thermal Prandtl Number, Pr ✓ Magnetic Prandtl Number, Pm
• To reproduce “Experiments on Rayleigh-Benard convection, magnetoconvection and rotating magnetoconvection in liquid gallium”, J. M. AURNOUy AND P. L. OLSON, J. Fluid Mech. (2001), vol. 430, pp. 283-307. 5 Physical Model A horizontally stratified fluid layer of characteristic height 푑, Cartesian coordinate system 푦 axis pointing vertically upward 푥, 푧 axes in the horizontal direction for the three-dimensional model.
6 Liquid : Gallium Walls : Copper
Externally applied vertical Magnetic field
Top Isothermal Cold wall Conducting
Bottom Isothermal Hot wall Conducting
Side walls : insulated walls, horizontal periodic conditions
7 Physical properties of Gallium
Property Units Value Density, 휌 kg m-3 6.095 X 103
Melting temperature, T0 ͦC 29.7 Thermal expansion coefficient, 훽 K-1 1.27 X 10-4 -1 -1 Specific heat, CP J kg K 397.6 Kinematic viscosity, 휐 m2s-1 3.2 X 10-7 휅 2 -1 -5 Thermal diffusivity, 훼 = m s 1.27 X 10 휌퐶푃 Thermal conductivity, 휅 W m-1 K-1 31 Magnetic diffusivity, 휂 m2s-1 0.21 Electrical conductivity,휎 (ohm m)-1 3.85 X 106
Thermal Prandtl Number = Pr1 = = 0.025
Magnetic Prandtl Number = =1.5E-6
푃푟 Roberts Number = 2 = 0.0006 푃푟1 8 The considered RMC system is a straightforward extension of the Lorenz model for Boussinesq convection, with the Lorentz force taken into account. ഥ ഥ 휌 ′ ′ 훻. 푉′ = 0, 훻. 퐵′ = 0, = 1 − 훽 푇 − 푇 0 휌0
휕푉′ഥ 1 휌 ഥ 2 ′ + 푉ത′. 훻′ 푉′ = − 훻푝 − g푒ෞ푦 + 휈훻 푉 휕푡′ 휌0 휌0 1 + 훻 × 퐵′ × 퐵′ − Ωഥ × Ωഥ × 푟ҧ + 2 푉ത × Ωഥ 휇0휌0 Transient term + Advection (Nonlinear) term = Pressure force + buoyancy force + Lorenz force + Coriolis force +centrifugal force 휕퐵′ഥ = 훻 × 푉′ഥ × 퐵′ഥ + 휂훻2퐵ത′ 휕푡′
휕푇′ = −훻. 푇′푉′ + 훼훻2푇′ 휕푡′ Solutions at Conduction State
• 푉ത푠 = 0, ′ • 푇′푠 = 푇′0 − 훽ሶ푦 , where 훽ሶ is the adverse temperature gradient. 1 • 푝′ = 푝′ − 푔휌 푦′ + 훽훽ሶ푦푦′2 푠 0 0 2 ′ ′ • 퐵 푠 = 퐵 0푒ෞ푦
Consider small perturbations in the equilibrium ҧ ҧ ഥ solutions as 푓 = 푓푠+푓∗, where f is the field variable.
10 Basic (Non-dimensional) equations
The quantities 푉∗′, 푇∗′,푡′, 푃∗′, 퐵ത′∗ are made dimensionless by using the scales ′ 2 −2 2 훼Τ푑 , Δ푇 , 푑 Τ훼 , 휌0훼 Τ푑 , 훼퐵′0Τ휂
훻. 푉ത = 0, 훻. 퐵ത = 0
1 휕푉ത 푃푟 + (푉ത. 훻)푉ത − 푄 2 퐵ത. 훻 퐵ത = 푃푟1 휕푡 푃푟1 ത 푃 푄 푃푟2 2 푇푎 푃푟1 2 휕퐵 = −훻 + 퐵ത + 푄퐵푦 − 푒ෞ푦 × 푟ҧ + 푄 푃푟1 2 푃 푟1 8 휕푧
1/2 2 +푅푎 푇푒ෞ푦 + 푇푎 푉ത × 푒ෞ푦 + 훻 푉ത
휕 − 훻2 푇 = 푤 − 푉ത. 훻 푇 휕푡
푃푟2 휕 2 푃푟2 − 훻 퐵ത = 훻 × 푉ത × 푒ෞ푦 + 훻 × 푉ത × 퐵ത 푃푟1 휕푡 푃푟1
11 Non-dimensional numbers
• Rayleigh number, 𝑔 훽 ∆푇 푑3 Buoyancy force Ra = = 훼휈 visocous force • Chandrasekhar number, 2 2 휇 퐵 푑 LorenzLorentz forceforce 푄 = 0 0 = 휌0휈휂 viscous force 4 Ω2푑4 CCoriolisoriolos forceforce • Taylor number, 푇푎 = = 휈2 Viscous force
ℎ 푑 • Nusselt number, 푁푢 = 휅 Δ푇
12 Methodology • Mesh : Uniform (Nodes : 80864 )
• Finite Volume method • Advection terms: First order upwind • Pressure : SIMPLE (Semi Implicit Pressure Linked Equations, Patankar 1979) • Temperature : Second order upwind • Magnetic field : First order upwind • System of algebraic equations – AMG (Algebraic Multi Grid) Solver • Convergence criteria < 1E-4
13 RBC 2D Q = 0, Ta = 0
Note: Temperature is given in Kelvin and velocity is given in m/s.
14 Isotherms
When Ra < 1708 we get Nu < 1, i.e., subcrtical flow. Heat propagates in the system as a conduction mode.
15 Isotherms
For Ra > 1708 we get supercritical flow.
Theoretically Chandrasekar (1961) showed that for RBC only Stationary convection occur as a first instability at the onset.
16 Streamlines
For Ra > 1708 we get symmetric rolls having y-axis as a axis of rotation.
17 As Ra increases horizontal length increases.
18 Experiments on Rayleigh-Benard convection, magnetoconvection and rotating magnetoconvection in liquid gallium. J. M. AURNOU AND P. L. OLSON. J. Fluid Mech. (2001), vol. 430, pp. 283-307.
20 RBC 3D Magnetoconvection (MC)
With Ta = 0
21 Weak Flow w(m/sec)
Cylindrical rolls with sneaking structures 22 Magnetoconvection = 1210, Ra = 27100 Nu = 1.02
Onset of convection delayed due to the effect of Lorenz force on thermal buoyancy force 23 Induced Vertical Magnetic field = 1210
Onset of convection delayed due to the effect of Lorenz force on thermal buoyancy force.
24 Temperature Variation = 1210
25 = 2000, Ra = 30139 Nu = 1.01 (Chandrasekhar (1961)
26 Plane layer Dynamo RBC with magnetic field and rotation
27 Ta = 970, Q = 1210 and Ra = 25500
28 Stress free boundary conditions Ta = 0, Q = 0 Aspect Ratio (=L/D)= 4 L Cold Wall d
Hot Wall 휆 ≅ 푑 2
2휋 Diameter of the convection cells is half the critical wave-length 휆푐 = = 2.82 , 푘푐 푘푐 is the critical wavenumber and 푘푐 = 휋/ 2.
Critical Ra = 657.25 29 Comparison 0.4
Weakly Non-linear Analysis DNS
0.2 Log(Nu)
0.0
2.80 2.84 2.88 2.92 2.96 3.00 Log(Ra)
Weakly non-linear analysis – Amplitudes 퐴𝑖, 푖 = 1, 2, 3, 4, 5, 6, [Kuo (1961)]
30 Ra = 658
Ra = 758 Streamlines
Ra = 893
Ra = 898 Isotherms Ra = 658
Ra = 758
Ra = 893
Ra = 1183 Ra = 658 Velocity magnitude
Ra = 758
Ra = 893
Ra = 898 Conclusions • In RBC when Ra < 1708 we get Nu < 1, i.e. the existence of the sub- critical flow. This result implies that heat can propagate in the system due to a “conductive mode”. • When Ra>1708 symmetric rolls perpendicular to vertical Y -axis are observed. • As Ra further increases from 1708, the horizontal width of the rolls increases. • In the presence of an applied magnetic field, the MC shows the cylindrical rolls. For the Chandrasekhar number 2000, a unique pattern of 3D rectangular vortical structures is noticed. The onset of convection is delayed due to the effect of Lorentz force on thermal buoyancy force.
34 • In RMC at low rotation rates horizontally stretched multi-cellular rolls perpendicular to the gravity axis again arise. • Simulations show the oscillatory nature of thermal convection with the formation of thin Hartmann layers close to boundaries due to strong damping effect of the magnetic field on flow velocities and heat transfer. • For stress-free boundary conditions – Nu = 1.00 for Ra = 658 – the plumes increases in vertical direction with Ra.
35 References
• H.L. Kuo, Solution of the non-linear equations of the cellular convection and heat transport, J. Fluid Mech 10 (1961) 611-630. • J. M. Aurnou, P. L. Olson, Experiments on Rayleigh-Benard convection, magnetoconvection and rotating magnetoconvection in liquid gallium, J. Fluid Mech 430 (2001) 283-307. • Y.Rameshwar, M.A.Rawoof Sayeed, H.P.Rani, D.Laroze, Finite amplitude cellular convection under the influence of a vertical magnetic field, Int. J. Heat and Mass Transfer 114 (2017) 559-577. • S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, (1961) Oxford University Press.
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