脈衝電漿推力器磁流體在震波管的數值解

Numerical Solutions of the Shock-Tube Problem for the Plasma Flow in Pulsed Plasma Thruster

電漿推進的數值模擬是脈衝電漿推力器（PPT）基本研究的基礎。本研究通過理論推導和數值計算，發展了脈衝電漿推力器的磁流體流場模式。學理上基於馬克士威方程組（Maxwell's equations）和勞侖茲力（Lorentz force）方程式，結合流體守恆方程式成為理想化磁流體動力學（Magnetohydrodynamics, MHD）一維數學模型。本研究將對此MHD方程式組應用於震波管所形成的李曼問題（Riemann problem）進行數值解，本文採取三種數值方法做比較。三種方法雖然都得到相同的數值解，但是Lax-Wendroff和MacCormack methods不論所選擇的人工黏滯係數（artificial viscosity）大小均無法解決在不連續區數值振盪的問題。在考慮物理模式中波傳遞的加成作用，第三種數值方法採用Roe's Riemann solver，先對MHD方程組進行了特徵分析，推導其特徵矩陣，雖然過程比較複雜，但其解可以避免不連續區數值振盪的問題。本研究也採用高準確度方法解析不連續區域變數的變化。在時間積分方面採取四階Runge-Kutta方法積分，所得到數值結果與文獻上的結果符合。擴散波（rarefaction fan），接觸不連續（contact discontinuity）波，與震波（shock wave）都能得到清楚的解析。在磁流體力場中，壓力在接觸不連續的地方因為受到磁場的影響也呈現不連續的情況，必須定義一全壓包括磁場壓力如此定義的全壓跨過接觸不連續的是連續分布的。

Numerical simulation of the plasma propulsion is essential to the Pulsed Plasma Thruster (PPT) fundamental research. By way of coupling the effects among the electric field, magnetic field and flow field in the PPT, a one-dimensional time-dependent model based magnetohydrodynamic (MHD) for PPT has been developed and solved numerically. The system of idealized MHD equations is a conservation system of hydrodynamics and Maxwell's equations with Lorentz force as the driving force of the plasma flow. The shock-tube problem has been used to test numerical stabilities of employed numerical methods. In this research the Riemann problem for a quasilinear hyperbolic system of equations governing the one-dimensional unsteady flow of an inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field, is solved numerically. Three different numerical schemes have been employed in this study. The Lax-Wendroff and MacCormack were originally employed for the space discretization of the system of nonlinear equations. Unfortunately, the results showed oscillations at discontinuities even with addition of artificial viscosities for these two methods. For numerical simulations to be effective, a high-order accurate solver that captures MHD shocks monotonically and works reliably for strong magnetic fields is needed. A characteristics-based scheme of Roe's Riemann solver for the MHD equations, with flux limiters to improve spatial accuracy has been developed successfully. The unsteady terms are discretized using fourth order Runge-Kutta integration. The results are compared with other references for validation and show a good agreement with the existed literature. Rarefaction fan, contact discontinuity and a moving shock have been resolved clearly. It has been observed that in contrast to the gasdynamic case, the pressure varies across the contact discontinuity. A full pressure has been introduced to remove the spike caused by the influence of the magnetic field and is continuous across the contact discontinuity.