MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.     

Heating of the solar corona

We review a number of models which are currently being considered for coronal heating, but we consider also heating of the chromosphere which requires nearly as much energy as the active corona, and more energy than coronal holes or the quiet corona. There are basically two types of models, which are motivated by a variety of observations. (1) Models which invoke MHD waves generated by the convective motions are motivated by observations of the ubiquitous presence of Alfvén waves in the solar wind. There is evidence that these waves heat and accelerate the solar wind protons and heavy ions. The solar wind thus provides one example of wave heating. Waves have the advantage of being able to heat the chromosphere and photospheric magnetic flux tubes on their way to the corona. MHD turbulence (as observed in the solar wind) or resonance absorption seem to provide adequate dissipation mechanisms. A problem with wave theories is that the waves tend to be reflected by the steep Alfén speed gradient in the chromosphere and transition region, but it is estimated that adequate energy fluxes can enter the open corona, or closed coronal loops if global loop resonances can be excited. Short coronal loops (L≲10⁴ km) can also receive adequate wave energy fluxes even if the loop resonances are not excited, but a problem exists with getting enough energy into intermediate length loops (L≈10⁴-5×10⁴ km) since their resonant frequencies are possibly to high to be excited. (2) Models which invoke the gradual buildup of coronal magnetic energy due to random walks of the photospheric flux tubes, and the subsequent release of that energy via current sheet information and reconnection, are supported by observations indicating that localized impulsive heating and dynamic events occur in the transition region and corona. These models cannot explain the chromospheric heating or the coronal heating on open field lines. They require substantial random walks of the photospheric footpoints, which still need to observationally verified. A third possibility, which has not been studied in detail, is that the chromospheric and coronal heating is associated with emergence and cancellation of magnetic flux. All types of models are ripe for further studies using numerical simulations, and along the way we shall offer several suggestions for fruitful numerical studies.     

When MHD-based microfluidics is equivalent to pressure-driven flow

Magneto-hydrodynamics (MHD) provides a convenient, programmable means for propelling liquids and controlling fluid flow in microfluidic devices without a need for mechanical pumps and valves. When the magnetic field is uniform and the electric field in the electrolyte solution is confined to a plane that is perpendicular to the direction of the magnetic field, the Lorentz body force is irrotational and one can define a “Lorentz” potential. Since the MHD-induced flow field under these circumstances is identical to that of pressure-driven flow, one can utilize the large available body of knowledge about pressure-driven flows to predict MHD flows and infer MHD flow patterns. In this note, we prove the equivalence between MHD flows and pressure-driven flows under certain conditions other than flow in straight conduits with rectangular cross sections. We determine the velocity profile and the efficiency of MHD pumps, accounting for current transport in the electrolyte solutions. Then, we demonstrate how data available for pressure-driven flow can be utilized to study various MHD flows, in particular, in a conduit patterned with pillars such as may be useful for liquid chromatography and chemical reactors. In addition, we examine the effect of interior obstacles on the electric current flow in the conduit and show the existence of a particular pillar geometry that maximizes the current.     

Theoretical analysis of the frictional losses in magnetohydrodynamic microflows considering slippage

In this article, we study the frictional losses in magnetohydrodynamic (MHD) microflows by analyzing the Poiseuille number defined through the Darcy–Weisbach friction factor. We consider two-dimensional fully developed flow models characteristic of MHD micropumps including the Hartmann braking effect and the existence of slippage. Unlike the purely hydrodynamic case, in MHD flows the Poiseuille number depends not only on the aspect ratio but also on the physical properties of the fluid and the externally applied magnetic field. Three different combinations of boundary conditions (slip and no-slip) are investigated. Calculations show that the Poiseuille number is considerably reduced as the dimensionless slip length is increased, while it increases as Hartmann number does. The obtained results are consistent with previous models and are helpful for the design of magnetohydrodynamic microflow devices.

     

The impacts of the ALE and hydrostatic-pressure approaches on the energy budget of unsteady free-surface flows

This paper focuses on the energy budget in the calculation of unsteady free-surface flows on moving grids with and without using the ‘arbitrary Lagrangian-Eulerian’ (ALE) formulation or hydrostatic-pressure assumption. The numerical tool is an in-house general-purpose solver for the unsteady, incompressible and homogeneous Navier-Stokes equations in a Cartesian domain. An explicit fractional-step method and co-located finite-volume method are used for the second-order accurate integrations in time and space. The test cases are nonlinear and linear irrotational standing waves, which allow to characterise the impacts of an ALE or Eulerian formulation with moving grids by comparison with the anticipated energy conservation. The study is also extended to viscous waves for varying wave-height-to-water-depth and basin aspect ratios. The Eulerian viewpoint produces marked overdamping as early as in the first wave period for the range of relative wave heights η₀/h > 0.01, where η₀ is the wave semi-amplitude and h is the undisturbed water depth. The hydrostatic calculations misrepresent the evolution of the potential and kinetic energies for h/L > 0.1, where L is the basin length, with spurious modes arising from different initial conditions.     

BOUT++: A framework for parallel plasma fluid simulations

A new modular code called BOUT++ is presented, which simulates 3D fluid equations in curvilinear coordinates. Although aimed at simulating Edge Localised Modes (ELMs) in tokamak x-point geometry, the code is able to simulate a wide range of fluid models (magnetised and unmagnetised) involving an arbitrary number of scalar and vector fields, in a wide range of geometries. Time evolution is fully implicit, and 3rd-order WENO schemes are implemented. Benchmarks are presented for linear and non-linear problems (the Orszag-Tang vortex) showing good agreement. Performance of the code is tested by scaling with problem size and processor number, showing efficient scaling to thousands of processors.Linear initial-value simulations of ELMs using reduced ideal MHD are presented, and the results compared to the ELITE linear MHD eigenvalue code. The resulting mode-structures and growth-rate are found to be in good agreement (γBOUT++=0.245ωA, γELITE=0.239ωA, with Alfvénic timescale 1/ωA=R/VA). To our knowledge, this is the first time dissipationless, initial-value simulations of ELMs have been successfully demonstrated.     

Magnetohydrodynamic state estimation with boundary sensors

We present a PDE observer that estimates the velocity, pressure, electric potential and current fields in a magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow. This flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier-Stokes equations and a Poisson equation for the electric potential under the so-called inductionless MHD approximation in a low magnetic Reynolds number regime. We identify physical quantities (measurable on the wall of the channel) that are sufficient to generate convergent estimates of the velocity, pressure, and electric potential field away from the walls. Our observer consists of a copy of the linearized MHD equations, combined with linear injection of output estimation error, with observer gains designed using backstepping. Pressure, skin friction and current measurements from one of the walls are used for output injection. For zero magnetic field or nonconducting fluid, the design reduces to an observer for the Navier-Stokes Poiseuille flow, a benchmark for flow control and turbulence estimation. We show that for the linearized MHD model the estimation error converges to zero in the L² norm. Despite being a subject of practical interest, the problem of observer design for nondiscretized 3-D MHD or Navier-Stokes channel flow has so far been an open problem.     

Numerical experimentation on spherically symmetric one-dimensional magnetohydrodynamic (MHD) wave propagation

Radial propagation of one-dimensional magnetohydrodynamic (MHD) waves are analyzed numerically on the basis of the Implicit-Continuous-Fluid-Eulerian (ICE) scheme. Accuracy of the numerical method and other properties are tested through the study of MHD wave propagation. The three different modes of MHD waves (i.e. fast- slow- and Alfven (transverse) mode) are generated by applying physically consistent boundary perturbations derived from MHD compatibility relations. It is shown that the resulting flow following these waves depend upon the relative configurations of the initial magnetic field and boundary perturbations.     

The modified differential transform method for solving MHD boundary-layer equations

A new analytical method (DTM-Padé) was developed for solving magnetohydrodynamic boundary-layer equations. It was shown that differential transform method (DTM) solutions are only valid for small values of independent variable. Therefore the DTM is not applicable for solving MHD boundary-layer equations, because in the boundary-layer problem y→∞. Numerical comparisons between the DTM-Padé and numerical methods (by using a fourth-order Runge-Kutta and shooting method) revealed that the new technique is a powerful method for solving MHD boundary-layer equations.     

Numerical Analysis of an Artificial Compression Method for Magnetohydrodynamic Flows at Low Magnetic Reynolds Numbers

Magnetohydrodynamics (MHD) is the study of the interaction of electrically conducting fluids in the presence of magnetic fields. MHD applications require substantially more efficient numerical methods than currently exist. In this paper, we construct two decoupled methods based on the artificial compression method (uncoupling the pressure and velocity) and partitioned method (uncoupling the velocity and electric potential) for magnetohydrodynamics flows at low magnetic Reynolds numbers. The methods we study allow us at each time step to solve linear problems, uncoupled by physical processes, per time step, which can greatly improve the computational efficiency. This paper gives the stability and error analysis, presents a brief analysis of the non-physical acoustic waves generated, and provides computational tests to support the theory.     

An Optimized Low-Dissipation Monotonicity-Preserving Scheme for Numerical Simulations of High-Speed Turbulent Flows

This paper presents an optimized low-dissipation monotonicity-preserving (MP-LD) scheme for numerical simulations of high-speed turbulent flows with shock waves. By using the bandwidth dissipation optimization method (BDOM), the linear dissipation of the original MP scheme of Suresh and Huynh (J. Comput. Phys. 136, 83–99, 1997) is significantly reduced in the newly developed MP-LD scheme. Meanwhile, to reduce the nonlinear dissipation and errors, the shock sensor of Ducros et al. (J. Comput. Phys. 152, 517–549, 1999) is adopted to avoid the activation of the MP limiter in regions away from shock waves. Simulations of turbulent flows with and without shock waves indicate that, in comparison with the original MP scheme, the MP-LD scheme has the same capability in capturing shock waves but a better performance in resolving small-scale turbulence fluctuations without introducing excessive numerical dissipation, which implies the MP-LD scheme is a valuable tool for the direct numerical simulation and large eddy simulation of high-speed turbulent flows with shock waves.     

A global collisionless PIC code in magnetic coordinates

A global plasma turbulence simulation code, ORB5, is presented. It solves the gyrokinetic electrostatic equations including zonal flows in axisymmetric magnetic geometry. The present version of the code assumes a Boltzmann electron response on magnetic surfaces. It uses a Particle-In-Cell (PIC), δf scheme, 3D cubic B-splines finite elements for the field solver and several numerical noise reduction techniques. A particular feature is the use of straight-field-line magnetic coordinates and a field-aligned Fourier filtering technique that dramatically improves the performance of the code in terms of both the numerical noise reduction and the maximum time step allowed. Another feature is the capability to treat arbitrary axisymmetric ideal MHD equilibrium configurations. The code is heavily parallelized, with scalability demonstrated up to 4096 processors and 10⁹ marker particles. Various numerical convergence tests are performed. The code is validated against an analytical theory of zonal flow residual, geodesic acoustic oscillations and damping, and against other codes for a selection of linear and nonlinear tests.     

Turbulence models and their applications to the prediction of internal flows: A review

The paper presents a brief account of various turbulence models employed in the computation of turbulent flows, and evaluates the application of these models to internal flows by examining the predictions of various turbulence models in selected important flow configurations. The main conclusions of this analysis are: (a) The κ-ε model is used in a majority of all the 2-D flow calculations reported in the literature. (b) Modified forms of the κ-ε model improve the performance for flows with streamline curvature and heat transfer. (c) For flows with swirl, the κ-ε model performs rather poorly; the algebraic stress model performs better in this case. (d) For flows with regions of secondary flow (noncircular duct flows), the algebraic stress model performs fairly well. Two important factors in the numerical solution of the model equations, namely false diffusion and inlet boundary conditions, are discussed. The existence of countergradient transport and its implications in turbulence modeling are examined. Finally, some recommendations for improving the model performance are made. The need for detailed experimental data in flows with strong curvature is emphasized.     

Adaptive filtering and limiting in compact high order methods for multiscale gas dynamics and MHD systems

The adaptive multistep linear and nonlinear filters for multiscale shock/turbulence gas dynamics and magnetohydrodynamics (MHD) flows of the authors are extended to include compact high order central differencing as the spatial base scheme. The adaptive mechanism makes used of multiresolution wavelet decomposition of the computed flow data as sensors for numerical dissipative control. The objective is to expand the work initiated in [Yee HC, Sjögreen B. Nonlinear filtering in compact high order schemes. In: Proceedings of the 19th ICNSP and 7th APPTC conference; 2005; J Plasma Phys 2006;72:833–36] and compare the performance of adaptive multistep filtering in compact high order schemes with adaptive filtering in standard central (non-compact) schemes for multiscale problems containing shock waves.     

One-step minimum Hellinger distance estimation

It is well known now that the minimum Hellinger distance estimation approach introduced by Beran (Beran, R., 1977. Minimum Hellinger distance estimators for parametric models. Ann. Statist. 5, 445-463) produces estimators that achieve efficiency at the model density and simultaneously have excellent robustness properties. However, computational difficulties and algorithmic convergence problems associated with this method have hampered its application in practice, particularly when the method is applied to models with high-dimensional parameter spaces. A one-step minimum Hellinger distance (MHD) procedure is investigated in this paper to overcome computational drawbacks of the fully iterative MHD method. The idea is to start with an initial estimator, and then iterate the Newton-Raphson equation once related to the Hellinger distance. The resulting estimator can be considered a one-step MHD estimator. We show that the proposed one-step MHD estimator has the same asymptotic behavior as the MHD estimator, as long as the initial estimators are reasonably good. Furthermore, our theoretical and numerical studies also demonstrate that the proposed one-step MHD estimator also retains excellent robustness properties of the MHD estimators. A real data example is analyzed as well.     

Particle simulation methods in plasma physics

A brief survey of particle simulation methods is presented. Section 2 outlines the particle-mesh (PM) method for collisionless phase fluids. Section 3 describes how PM is extended to make simulation studies of dense plasmas feasible: The particle-particle/particle-mesh (P³M) algorithms reduces the cost scaling to αN_p from the αN²_p for conventional methods, where N_p is the number of particles. Section 4 summarises particle modelling of fluids and magnetofluids. New algorithms (EPIC schemes) arise when the optimisation techniques of collisionless PM models are applied to the fluid and MHD equations. These are shown to outperform state-of-the-art finite difference schemes in both cost and performance.     

Numerical study of an explosion in a non-homogeneous medium with and without magnetic fields

A version of the two step Lax-Wendroff difference method with second order accuracy is used to seek solutions of the unsteady magnetohydrodynamic (MHD) equationsfor the study of an explosion in a non-homogeneous medium with and without magnetic fields. The explosion is initiated by introducing a finite amount of energy in a small volume of gas which leads to an instantaneous increase of gas temperature. Two types of magnetic field configurations (i.e. open and closed) are considered to illustrate the dependence and differences of magnetic and a non-magnetic flow motion. Numerical results show the strong dependence of induced MHD flow field on the magnetic field configuration and strength. The development and decay of the fast and slow MHD shock waves, as well as the ordinary gasdynamic shock waves, are also represented well by the present computer simulations. In conclusion, we have demonstrated that the numerical scheme we have used is a reliable one for studying 2-dimensional MHD problems in non-homogeneous medium.     

Generic congestion control

The nowadays Internet architecture is mainly based on unicast communications and best-effort service. However, the development of the Internet encouraged emerging services that are sensitive to delay or packet loss, as it is the case for multimedia and group applications. The deployment of these applications should not compromise the proper transmission of TCP flows and would benefit significantly from flows that are responsive to congestion.We propose efficient congestion avoidance mechanism (ECAM)¹ as a generic framework for congestion control in the Internet, to address this lack and important need of congestion control in various situations that occurs in the Internet. ECAM is designed for uncontrolled unicast and multicast traffic and supports both reliable and unreliable best-effort flows. ECAM works not only for best-effort service, but supports as well the new differentiated services, where out of profile packets may experience congestion. Implementation problems are also discussed.     

Decomposition of group flows in regular matroids

In a regular matroidM=(E,C) we discuss two different approaches to group matroid flows. By showing that these two approaches are equivalent we set up a decomposition theory for group matroid flows and derive two algorithms for decomposing group matroid flows. The second one finds so-called positive decomposition, a fact which is highly important in applications. By specializing the results to graphic and co-graphic matroids we generalize some well known results of real-valued network flow theory to group network flows and derive some new results for tensions.