Approximation of the time-dependent induction equation with advection using Whitney elements

2016 ◽  
Vol 35 (1) ◽  
pp. 326-338 ◽  
Author(s):  
Caroline Nore ◽  
Houda Zaidi ◽  
Frederic Bouillault ◽  
Alain Bossavit ◽  
Jean-Luc Guermond
Keyword(s):  
Velocity Field ◽  
Dynamo Action ◽  
Content Type ◽  
Kinematic Dynamo ◽  
Von Karman ◽  
Whitney Forms ◽  
Dependent Variables ◽  
Induction Equation ◽  
Von Kármán ◽  
Good Agreement

Purpose – The purpose of this paper is to present a new formulation for taking into account the convective term due to an imposed velocity field in the induction equation in a code based on Whitney elements called DOLMEN. Different Whitney forms are used to approximate the dependent variables. The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. DOLMEN is developed to investigate the dynamo action in non-axisymmetric domains like the impeller driven flow of the von Kármán Sodium (VKS) experiment. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field. Design/methodology/approach – Different Whitney forms are used to approximate the dependent variables. The vector potential is discretized using first-order edge elements of the first family. The velocity is approximated by using the first-order Raviart-Thomas elements. The time stepping is done by using the Crank-Nicolson scheme. Findings – The authors study the kinematic dynamo action in a von Kármán configuration and obtain results in good agreement with those provided by another well validated code called SFEMaNS. The authors show that a 3D magnetic field dominated by an axisymmetric vertical dipole can grow in a kinematic dynamo configuration using an analytical velocity field. Originality/value – The findings offer a basis to a scenario for the VKS dynamo.

Kinematic dynamo simulations of von Kármán flows: application to the VKS experiment

2010 ◽  
Vol 74 (2) ◽  
pp. 165-176 ◽  
Author(s):  
A. Pinter ◽  
B. Dubrulle ◽  
F. Daviaud
Keyword(s):  
Kinematic Dynamo ◽  
Von Karman ◽  
Von Kármán

On the Velocity Distribution of Turbulent Flow in Pipes and Channels of Constant Cross Section

1946 ◽  
Vol 13 (2) ◽  
pp. A85-A90
Author(s):  
Chi-Teh Wang
Keyword(s):  
Turbulent Flow ◽  
Velocity Distribution ◽  
Length Distribution ◽  
Critical Examination ◽  
Large Reynolds Number ◽  
Good Picture ◽  
Von Karman ◽  
Von Kármán ◽  
New Formula ◽  
Good Agreement

Abstract This paper follows the Prandtl conception of momentum transport and gives a critical examination of the so-called Prandtl-Nikuradse formula and the von Kármán formula for the velocity distribution of the turbulent flow in tubes or channels at large Reynolds number. It shows that both formulas would not give a good picture of the turbulent flow near the center of the conduit, and indeed they actually do not. A new formula for the velocity distribution is developed from a study of the mixing-length distribution across the section. This new formula checks quite well with the experiments and yields the same skin-friction formula as derived by von Kármán and Prandtl, which itself is in very good agreement with experiments.

A computational investigation for propagation of elasto-viscoplastic zones in the shock loaded circular plates

2014 ◽  
Vol 31 (7) ◽  
pp. 1401-1443 ◽  
Author(s):  
Asghar Zajkani ◽  
Abolfazl Darvizeh ◽  
Mansour Darvizeh
Keyword(s):  
Shock Wave ◽  
Constitutive Equations ◽  
Time History ◽  
Structural Equations ◽  
Computational Time ◽  
Mapping Technique ◽  
Circular Plates ◽  
Content Type ◽  
Von Karman ◽  
Von Kármán

Purpose – The purpose of this paper is to introduce a computational time dependent modeling to investigate propagation of elastic-viscoplastic zones in the shock wave loaded circular plates. Design/methodology/approach – Constitutive equations are implemented incrementally by the Von-Kármán finite deflection system which is coupled with a mixed strain hardening rule and physical-base viscoplastic models. Time integrations of the equations are done by the return mapping technique through the cutting-plane algorithm. An integrated solution is established by pseudo-spectral collocation methodology. The Chebyshev basis functions are utilized to evaluate the coefficients of displacement fields. Temporal terms are discretized by the Houbolt marching method. Spatial linearizations are accomplished by the quadratic extrapolation technique. Findings – Results of the center point deflections, effective plastic strain and stress (dynamic flow stress) and temperature rise are compared for three features of the Von-Kármán system. Identifying time history of resultant stresses, propagations of the viscoplastic plastic zones are illustrated for two circumstances; with considering strain rate and hardening effects, and without them. Some of modeling and computation aspects are discussed, carefully. When the results are compared with experimental data of shock wave loadings and finite element simulations, good agreements between them are observed. Originality/value – This computational approach makes coupling the structural equations with the physical descriptions of the high rate deformation through step-by-step spectral solution of the constitutive equations.

A Born–von Karman Force Constant Model for Aluminum

1974 ◽  
Vol 52 (17) ◽  
pp. 1714-1715 ◽  
Author(s):  
E. R. Cowley
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Force Constant ◽  
Frequency Distribution ◽  
Brillouin Zone ◽  
Normal Modes ◽  
Frequency Distribution Function ◽  
Von Karman ◽  
Von Kármán ◽  
Good Agreement

A Born–von Karman force constant model of aluminum, fitted to the frequencies of normal modes with wave vectors distributed throughout the Brillouin zone, is described, and the frequency distribution function calculated. The result is in very good agreement with a distribution function calculated directly from the experimental data.

scholarly journals Numerical simulation of the von Kármán sodium dynamo experiment

2018 ◽  
Vol 854 ◽  
pp. 164-195 ◽  
Author(s):  
C. Nore ◽  
D. Castanon Quiroz ◽  
L. Cappanera ◽  
J.-L. Guermond
Keyword(s):  
Reynolds Number ◽  
Magnetic Permeability ◽  
Magnetic Reynolds Number ◽  
Liquid Sodium ◽  
Dynamo Action ◽  
Azimuthal Component ◽  
Eddy Simulation ◽  
Von Karman ◽  
Hydrodynamic Experiments ◽  
Von Kármán

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.

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