Upwind Numerical Method Research Articles

Linear simulations of the cylindrical Richtmyer-Meshkov instability in magnetohydrodynamics

Numerical simulations and analysis indicate that the Richtmyer-Meshkov instability (RMI) is suppressed in ideal magnetohydrodynamics (MHD) in Cartesian slab geometry. Motivated by the presence of hydrodynamic instabilities in inertial confinement fusion and suppression by means of a magnetic field, we investigate the RMI via linear MHD simulations in cylindrical geometry. The physical setup is that of a Chisnell-type converging shock interacting with a density interface with either axial or azimuthal (2D) perturbations. The linear stability is examined in the context of an initial value problem (with a time-varying base state) wherein the linearized ideal MHD equations are solved with an upwind numerical method. Linear simulations in the absence of a magnetic field indicate that RMI growth rate during the early time period is similar to that observed in Cartesian geometry. However, this RMI phase is short-lived and followed by a Rayleigh-Taylor instability phase with an accompanied exponential increase in the perturbation amplitude. We examine several strengths of the magnetic field (characterized by β=2pBr2) and observe a significant suppression of the instability for β ≤ 4. The suppression of the instability is attributed to the transport of vorticity away from the interface by Alfvén fronts.

An Artificial Viscosity for the III-Posed One-Dimensional Incompressible Two-Fluid Model

Methods to remedy the ill-posedness of the basic one-dimensional two-fluid model, which is widely used in nuclear reactor safety codes, have been the subject of considerable study. Both of the two prevalent methods have drawbacks. Unconditional hyperbolization uses nonphysical constitutive relations to create a well-posed two-fluid model that is hyperbolic over all flow conditions. However, when the model is hyperbolized, it is also stabilized, which is not a universal property of two-phase flows. The second method, the preferred method of the U.S. Nuclear Regulatory Commission safety codes, is to simply use a first-order upwind numerical method that relies on numerical viscosity to regularize the ill-posedness of the model by damping the short-wavelength instabilities. Unfortunately, the scale of the “short wavelength” is related to a particular numerical grid or discretization. Because of the consistency of the numerical method, in the limit of an infinitely resolved grid, i.e., Δx, Δt→0, the numerical viscosity vanishes, as does its regularization effect. This results in a somewhat heuristic user guideline that suggests a lower limit on the grid size based on a cross-sectional dimension that is a combination of the long-wavelength assumption and experience. However, a cutoff wavelength achieved by numerical viscosity is not set by the grid size alone but also depends on the time step, the material, and the flow properties, as demonstrated with a von Neumann stability analysis. This can create poor resolution in areas where numerical stability may not be a substantial problem, unless the guideline is intentionally violated. Additionally, strict observance of this limit makes verification by convergence difficult or impossible. Therefore, it is proposed that an artificial viscosity be prescribed explicitly, i.e., independently of any particular numerical method or grid. An artificial viscosity model is derived that prescribes exactly a cutoff in the linear stability growth rate at a specified wavelength, e.g., λ0 = 2DH consistent with the aforementioned user guideline. It is shown, using the water faucet problem, that the proposed artificial viscosity model can be used to remove the high-frequency component of the solution without limiting the resolution of the grid. Furthermore, the solution also converges, which was not the case without the artificial viscosity.

Godunov’s Method for Simulatinons of Fluid-Structure Interaction in Piping Systems

Constant coefficient one-dimensional linear hyperbolic systems of partial differential equations (PDEs) are often used for description of fluid-structure interaction (FSI) phenomena during transient conditions in piping systems. In the past, these systems of equations have been numerically solved with method of characteristics (MOCs). The MOC method is actually the most efficient and accurate method for description of the single-phase transient in the cold liquid where the constant coefficient mathematical model describes phenomenon with sufficient accuracy. In energy production systems where hot pressurized liquid is used for heat transfer between the heat source and the steam generator, more complex and nonlinear mathematical models are needed to describe transient flow and these models cannot be solved with MOC method because the models are not constant. In addition, the MOC method can be used for pipes having discontinuities like elbows, geometrical changes, material properties changes, etc., but only with some extra numerical modeling. An interesting alternative is explicit characteristic upwind numerical method, known as Godunov’s method that is frequently used for nonlinear systems or systems where properties change with position. In the present study, applicability of the Godunov’s method for the FSI analyses is tested with eight first order PDEs mathematical model. The conventional linear mathematical model is improved with convective term that makes the system nonlinear and additional terms that enable simulations of the FSI in arbitrarily shaped piping systems located in a plane. Two PDEs describe pressure waves in the single-phase fluid and six PDEs describe axial, lateral, and rotational stress waves in the pipe. The applied system of equations has stiff source terms. This numerical problem is solved introducing implicit iterations. The proposed model is verified with a rod impact experiment that is carried out on single-elbow pipe hanging on wires. Godunov’s method is found as a very promising numerical method for simulations of the FSI problems.

Characteristic development of hyperbolic two-dimensional two-fluid model for gas–liquid flows with surface tension

A new hyperbolic, two-dimensional two-fluid model is developed to properly solve two-phase gas–liquid flows. Adopting the interfacial pressure jump terms in the momentum equations, the numerical stability is confirmed owing to the improvement in the mathematical property of the equation system. The derivation of the interfacial pressure jump terms is based on the infinitesimal surface-tension effect incorporated in the pressure difference at the gas–liquid interface. Through the characteristic analysis on the equation system, the eight eigenvalues are obtained analytically and they are proved real values representing phasic convective velocities and phasic sound speeds. Furthermore, the characteristic sound speeds are comparable with the earlier experimental data in excellent agreements. In addition, the eigenvectors are obtained analytically and they are shown to be linearly independent. Consequently, the governing equation system is mathematically hyperbolic with reasonable characteristic speeds by which the upwind numerical method avails. Advantage and possibility of the present model are discussed in some detail.

A High-Resolution Numerical Method for a Two-Phase Model of Deflagration-to-Detonation Transition

A conservative, upwind numerical method is formulated for the solution of a two-phase (reactive solid and inert gas) model of deflagration-to-detonation transition (DDT) in granular energetic solids. The model, which is representative of most two-phase DDT models, accounts for complete nonequilibrium between phases and constitutes a nonstrictly hyperbolic system of equations having parabolic degeneracies. The numerical method is based on Godunov's methodology and utilizes a new approximate solution for the two-phase Riemann problem for arbitrary equations of state. The approximate solution is similar to the Roe-type Riemann solution for single-phase systems. The method is able to accurately capture strong shocks associated with each phase without excessive smearing or spurious oscillations and can accurately resolve fine-scale detonation structure resulting from interaction between phases. The utility of the method is demonstrated by comparing numerical predictions with known solutions for three test cases: (1) a two-phase shock tube problem; (2) the evolution of a steady compaction wave in a granular material resulting from weak piston impact (∼100 m/s); and (3) the evolution of a steady two-phase detonation wave in an energetic granular material resulting from weak piston impact. The nominally second-order accurate numerical method is shown to have global convergence rates of 1.001 and 1.670 for inert test cases with (case 1) and without (case 2) discontinuities, respectively. For the reactive test case having a discontinuity (case 3), a convergence rate of 1.834 was predicted for coarse grids that seemed to be approaching the expected value of unity with increasing resolution.

An Upwind Numerical Method for Two-Fluid Two-Phase Flow Models

An upwind numerical method for a six equation two-fluid model is presented based on a linearized Riemann solver. The construction of this approximate Riemann solver uses an extension of Roe’s method that has been successfully used to solve gas dynamics equations in aerodynamics problems. As far as the two-fluid model is hyperbolic, this numerical method seems very efficient for the numerical solution of two-phase flow problems. In practice, this new numerical method has proved to be stable on nonstaggered grids and capable of generating accurate nonoscillating solutions for two-phase flow calculations. The scheme was applied both to shock tube problems and to standard tests for two-fluid codes.