A strong conservative Riemann solver for the solution of the coupled maxwell and Navier–Stokes equations (original) (raw)
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A Strong Conservative Implicit Riemann Solver for Coupled Navier-Stokes and Full Maxwell Equations
The effects of electromagnetic wave propagation, charge separation and higher-frequency effects can play a significant role in a plasma. Although the magnetohydrodynamic (MHD) model is the incumbent approach for describing plasmas of engineering interest, this model is incapable of resolving these features. In this paper, we introduce a split-flux Roe scheme approach for solving the single-fluid plasma model that retains the full Maxwell equations. This model is capable of resolving the missing effects. The approach is implemented in a multidimensional implicit dualtime solver and is validated against one- and two-dimensional problems of magnetohydrodynamics and electromagnetic wave propagation.
An AUSM-based Algorithm for Solving the Coupled Navier-Stokes and Maxwell Equations
The AUSM family of schemes has been successful in simulating fluid dynamic and magnetohydrodynamic flows. In this paper, we extend the AUSM method to solve the Navier-Stokes equations coupled to the full Maxwell equations. This approach permits the inclusion of electromagnetic wave propagation, displacement current and charge separation e ects. Validation of the approach is presented for problems of electromagnetic wave propagation and magnetohydrodynamics, which demonstrates a robust, unified hyperbolic method for solving both the wave and di usion limits in the same computational domain.
A new fully coupled solution of the Navier-Stokes equations
International Journal for Numerical Methods in Fluids, 1994
A fully coupled method for the solution of incompressible Navier-Stokes equations is investigated here. It uses a fully implicit time discretization of momentum equations, the standard linearization of convective terms, a cell-centred colocated grid approach and a block-nanodiagonal structure of the matrix of nodal unknowns. The Method is specific in the interpolation used for the flux reconstruction problem, in the basis iterative method for the fully coupled system and in the acceleration means that control the global efficiency of the procedure. The performance of the method is discussed using lid-driven cavity problems, both for two and three-dimensional geometries, for steady and unsteady flows.
Solving the Euler and Navier–Stokes equations by the AMR–CESE method
Computers & Fluids, 2012
The application of the AMR-CESE method for solving the Euler and Navier-Stokes equations is presented. The method is a combination of the space-time conservation element and solution element (CESE) method and the adaptive mesh refinement (AMR) technique. Its implementation is based on modification of the original CESE method and utilization of the framework of a parallel-AMR package PARAMESH to manage the block-AMR grid system. Furthermore, a variable time step algorithm is introduced to realize adaptivity of the method in both space and time. A test suite of standard problems for Euler and Navier-Stokes flows are calculated and the results show high resolution, high efficiency and versatility of the method for both shock capture and resolving boundary layer.
Journal of Computational Physics, 2004
We present numerical schemes for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematic equations. The result is a class of extremely efficient Navier-Stokes solvers. Full time accuracy is achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we prove unconditional stability of the new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier-Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C 0 elements to implement.
Named after Claude-Louis Navier and George Gabriel Stokes, the Navier Stokes Equations are the fundamental governing equations to describe the motion of a viscous, heat conducting fluid substances. These equations are obtained by applying Newton’s Law of motion to a fluid element and are also called the momentum equation. Supplemented by the mass conservation equation, these equations are also referred to as energy equation or continuity equation. Navier Stokes equations assume that the stress tensor in the fluid element is the sum of a diffusing viscous term that is proportional to the gradient of velocity, plus a pressure term (Batchelor 2000). Navier Stokes equations have wide range of applications in both academic and economical benefits. They are used as the basic algorithms in computational tools to simulate ocean currents, fluid flow in pipes, airflow around a foil and model the weather (Dean 2012). A Beam-warming algorithm coupled with Euler/Navier-Stokes equations can be applied for simulation of a transonic viscous flow over wings and the design of aircrafts. They can also help with the design of cars, mathematical modelling of the arterial blood flow in human body, and the design of power stations (Batchelor 2000). Last but not least, Maxwell’s equations in conjunction with Navier-Stokes equations can be used to design and study magnetohydrodynamics. (Dean 2012) The aim of this essay is to initially describe the properties of the Navier-Stokes equations and then create a simple way to understand the main components of the equation by describing the mass conservation, momentum conservation and heat equations. The paper only focuses on the motion of incompressible fluids.
Efficient variable-coefficient finite-volume Stokes solvers, arXiv preprint arXiv:1308.4605
2013
Abstract. We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity-pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565–7595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solvi...
Advected Upstream Splitting Method for the Coupled Maxwell and Navier–Stokes Equations
A solution procedure for the fully coupled Navier–Stokes and Maxwell equations is described. The approach implements a conservative fluid formulation in which the Lorentz body force and Ohmic heating terms are recast as convective terms. This removes explicit sources from the fluid equations, which have previously introduced severe stiffness and demanded a very delicate numerical treatment. The coupling with the full Maxwell equations enables displacement current effects, charge separation effects, low-conductivity plasma behavior, and electromagnetic wave propagation to be incorporated directly. To circumvent the issue of complicated eigenvectors, an AUSM-type flux splitting scheme is proposed. Validation of the approach is presented for problems of electromagnetic wave propagation in low-conductivity plasma and high-conductivity magnetohydrodynamic problems, which demonstrates a robust, unified hyperbolic method for resolving both the wave and diffusion limits of the electromagnetic behavior in the plasma.