A Strong Conservative Implicit Riemann Solver for Coupled Navier-Stokes and Full Maxwell Equations (original) (raw)
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We present a Finite Volume scheme for solving Maxwell's equations coupled to magnetized multi-fluid plasma equations for reactive and collisional partially ionized flows on unstructured meshes. The inclusion of the displacement current allows for studying electromagnetic wave propagation in a plasma as well as charge separation effects beyond the standard magnetohydrodynamics (MHD) description, however, it leads to a very stiff system with characteristic velocities ranging from the speed of sound of the fluids up to the speed of light. In order to control the fulfillment of the elliptical constraints of the Maxwell's equations, we use the hyperbolic divergence cleaning method. In this paper, we extend the latter method applying the CIR scheme with scaled numerical diffusion in order to balance those terms with the Maxwell flux vectors. For the fluids, we generalize the AUSM +-up to multiple fluids of different species within the plasma. The fully implicit second-order method is first verified on the Hartmann flow (including comparison with its analytical solution), two ideal MHD cases with strong shocks, namely, Orszag-Tang and the MHD rotor, then validated on a much more challenging case, representing a twofluid magnetic reconnection under solar chromospheric conditions. For the latter case, a comparison with pioneering results available in literature is provided.
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A solution procedure for the Navier-Stokes equations coupled with the full Maxwell equations is described. The approach implements a strongly conservative fluid formulation in which the Lorentz force and Ohmic heating terms are recast as convective terms. This removes explicit sources from the Navier-Stokes equations, which have previously introduced severe sti ness and demanded a very delicate numerical treatment. The coupling with the full Maxwell equations enables the displacement current to be incorporated directly. To demonstrate the e ectiveness of this technique, a fully explicit finite volume approximate Riemann solver is used to obtain numerical solutions to the Brio and Wu electromagnetic shock problem. Comparisons with the analytical solution show good agreement, and the implementation requires less than an hour of computational time on a single processor machine. Simulations using large and small conductivities confirm that the formulation captures both wave and di usion limits of the magnetic field.
Magnetohydrodynamics with Implicit Plasma Simulation
We consider whether implicit simulation techniques canbe extended in time and space scales to magnetohydrodynamics without any change but the addition of collisions. Our goal is to couple fluid and kinetic models together for application to multi-scale problems. Within a simulation framework, transition from one model to the other would occur not by a change of algorithm, but by a change of parameters. This would greatly simplify the coupling. Along the way, we have found new ways to impose consistent boundary conditions for the field solver that result in charge and energy conservation, and establish that numerically-generated stochastic heating is the problem to overcome. For an MHD-like problem, collisions are clearly necessary to reduce the stochastic heating. Without collisions, the heating rate is unacceptable. With collisions, the heating rate is significantly reduced.
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Despite its success at simulating accurately both non-neutral and quasi-neutral weakly-ionized plasmas, the drift-diffusion model has been observed to be a particularly stiff set of equations. Recently, it was demonstrated that the stiffness of the system could be relieved by rewriting the equations such that the potential is obtained from Ohm's law rather than Gauss's law while adding some source terms to the ion transport equation to ensure that Gauss's law is satisfied in non-neutral regions. Although the latter was applicable to multicomponent and multidimensional plasmas, it could not be used for plasmas in which the magnetic field was significant. This paper hence proposes a new computationally-efficient set of electron and ion transport equations that can be used not only for a plasma with multiple types of positive and negative ions, but also for a plasma in magnetic field. Because the proposed set of equations is obtained from the same physical model as the conventional drift-diffusion equations without introducing new assumptions or simplifications, it results in the same exact solution when the grid is refined sufficiently while being more computationally efficient: not only is the proposed approach considerably less stiff and hence requires fewer iterations to reach convergence but it yields a converged solution that exhibits a significantly higher resolution. The combined faster convergence and higher resolution is shown to result in a hundredfold increase in computational efficiency for some typical steady and unsteady plasma problems including non-neutral cathode and anode sheaths as well as quasi-neutral regions.
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As demonstrated by Parent, B., et al., (“Electron and Ion Transport Equations in Computational Weakly-Ionized Plasmadynamics,” Journal of Computational Physics, Vol. 259, 2014, pp. 51–69), the computational efficiency of the drift-diffusion plasma model can be increased significantly by recasting the equations such that the potential is obtained from Ohm’s law rather than Gauss’s law and by adding source terms to the ion transport equations to ensure that Gauss’s law is satisfied. Not only did doing so reduce the stiffness of the system, leading to faster convergence, but it also resulted in a higher resolution of the converged solution. The combined gains in convergence acceleration and resolution amounted to a hundredfold increase in computational efficiency when simulating nonneutral plasmas with significant quasi-neutral regions. In this paper, it is shown that such a recast of the drift-diffusion model has yet another advantage: its lack of stiffness permits the electron and ion transport equations to be integrated in coupled form along with the Favre-averaged Navier–Stokes equations. Test cases relevant to plasma aerodynamics (including nonneutral sheaths, magnetic field effects, and negative ions) demonstrate that the proposed coupled system of equations can be converged in essentially the same number of iterations as that describing nonionized flows while not sacrificing the generality of the drift-diffusion model.