Matching of asymptotic expansions of solutions of boundary value problems

We deal with the coupling of hyperbolic and parabolic systems in a domain Ω divided into two disjoint subdomains Ω+ and Ω− . Our main concern is to find out the proper interface conditions to be fulfilled at the surface separating the two domains. Next, we will use them in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, that is, the hyperbolic system is rendered parabolic by adding a small artificial “viscosity”. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These we take as interface conditions for the hyperbolic-parabolic problem. Actually, we discuss two alternative sets of interface conditions according to whether the regularization procedure is variational or nonvariational. We show how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, we discuss a method of resolution which alternates the resolution of the hyperbolic problem within Ω− and of the parabolic one within Ω+ . The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows.

On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach

The plasma flow in magnetoplasmadynamic (MPD) self-field thrusters is described by conservation equations for heavy particles, turbulence, electrons, and the magnetic field for reaction and thermal nonequilibrium. The equations are discretized on unstructured adaptive meshes. The numerical results, which are verified by experimental data, show that the newly developed finite volume code predicts the thrust well. It is found that electron pressure diffusion drives the arc out of nozzle-type MPD thrusters. The drop of density in front of a water-cooled anode is the reason for the beginning of thruster instabilities at high electric currents.

Numerical and Experimental Investigation of the Current Distribution in Self-Field Magnetoplasmadynamic Thrusters

Since the early eighties the development of plasma thrusters has been an important research topic at the IRS, where a broad spectrum of stationary plasma thrusters has been investigated experimentally as well as theoretically. High power MPD thrusters (50 kW-1 MW) and a wide range of thermal arcjets (0.5–150 kW) are developed under contracts with ESA, NASA, USAF, USNAVY and with German funding. The IRS has excellent installations for continuous tests of high power accelerators. The high current power supply is a current regulated d.c. thyristor rectifier of 6 MW. The maximum current is 48 kA with a ripple less than 1%. The vacuum system is a roots pump system consisting of four stages with a total suction power of about 250,000 m 3 /h at 1 Pa. Eight vacuum tanks of different sizes are connected to this system; six of them are used for plasma thruster development and two serve as plasma wind tunnels. For low power arcjet development and basic cathode erosion experiments four additional independent test stands are available.

Plasma thruster development program at the IRS

In a propulsion system, an electrohydrodynamic (EHD) body force is used to control the flow of a propellant through a micro channel, expansion slot, plenum chamber, or other flow region and thereby increase the specific impulse created by a propulsion system. In an embodiment, a plurality of electrodes are arranged and powered to create a plasma discharge, which can impart an EHD body force to a fluid. Various configurations of electrodes can be used to control the flow of the fluid into, out of, or through the flow region. In an embodiment, the use of EHD body forces can reduce, or substantially eliminate, shear forces on the surface of a plenum chamber, micro channel, or expansion slot of the propulsion system, resulting in a smooth flow of the propellant and increased thrust.

Method and apparatus for small satellite propulsion

We present nonoverlapping domain decomposition methods for the numerical treatment of compressible viscous plasma flows inside a self—field magnetoplasmadynamic (MPD) accelerator. The magneto—plasma flow is modelled by a system of conservation laws extended by partial differential equations describing the electromagnetic field. Since the numerical solution of the governing equations is extremely expensive, the flow—field domain is decomposed into two or three model zones, characterized by different physical properties of the flow. The compressible Navier—Stokes equations (extended under the influence of an arc discharge) in the near field of the accelerator are coupled with simplified models of extended conservation laws in the far field via appropriate transmission conditions at the artificial coupling boundaries. Results of our numerical computations are presented.

Heterogeneous Domain Decomposition Methods for Compressible Magneto-plasma Flows

We consider a two--dimensional coupled
transmission problem with the conservation laws for compressible
viscous flows, where in a subdomain $\Omega_1$ of the flow--field
domain $\Omega$ the coefficients modelling the viscosity and
heat conductivity are set equal to a small parameter
$\varepsilon>0$. The viscous/viscous coupled problem, say
$P_\varepsilon$, is equipped with specific boundary conditions
and natural transmission conditions at the artificial
interface $\Gamma$ separating $\Omega_1$ and $\Omega \setminus
\Omega_1$. Here we choose $\Gamma $ to be a line segment. The
solution of $P_\varepsilon$ can be viewed as a candidate for the
approximation of the solution of the real physical problem for
which the dissipative terms are strongly dominated by the
convective part in $\Omega_1$. With respect to the norm of uniform
convergence, $P_\varepsilon$ is in general a singular
perturbation problem. Following the Vishik--Ljusternik method, we
investigate here the boundary layer phenomenon at $\Gamma$. We
represent the solution of $P_\varepsilon$ as an asymptotic expansion of
order zero, including a boundary layer correction. We can show
that the first term of the regular series satisfies a reduced
problem, say $P_0$, which includes the inviscid/viscous
conservation laws, the same initial conditions as
$P_\varepsilon$, specific inviscid/viscous boundary conditions,
and transmission conditions expressing the continuity of the
normal flux at $\Gamma$. A detailed analysis of the problem for
the vector--valued boundary layer correction indicates whether
additional local continuity conditions at $\Gamma$ are necessary
for $P_0$, defining herewith the reduced coupled problem
completely. In addition, the solution of $P_0$ (which can be
computed numerically) plus the boundary layer correction at
$\Gamma$ (if any) provides an approximation of the solution of
$P_\varepsilon$ and, hence, of the physical solution as well. In
our asymptotic analysis we mainly use formal arguments, but we are
able to develop a rigorous analysis for the separate problem
defining the correctors.
Numerical results are in agreement with our asymptotic analysis.

Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows

For the numerical simulation of magnetoplasmadynamic (MPD) self‐field thruster flow, the solution of one of the two dynamical Maxwell equations – Faraday's law – is required. The Maxwell equations and Ohm's law for plasmas can be summarized in one equation for the stream function so that the two‐dimensional, axisymmetric magnetic field can be calculated. The finite volume (FV) discretization of the equation on unstructured, adaptive meshes is presented in detail and solutions for different thruster currents are shown. The calculated thrust is compared with the experimental data.

Discretization of the magnetic field in MPD thrusters

We are concerned with the mathematical modelling of high-enthalpy compressible plasma flows of magnetoplasmadynamic (MPD) rocket thrusters. A closed system of partial differential equations is associated with corresponding two-fluid flows which are subjected to strong electromagnetic fields. The numerical solution of the full original model requires nowadays huge computational costs. Therefore we use a heterogeneous domain decomposition where the corresponding coupled model is based on simplified conservation laws in a large subdomain. We establish new transmission conditions for the coupled MPD model, which appear as a byproduct of a detailed asymptotic analysis of the artificial interfaces that arise by domain decomposition. Moreover, a complete coupled mathematical MPD model is formulated, including appropriate outer boundary conditions. The well-posedness of this model is discussed by using mathematical as well as physical arguments.

On some mathematical modelling of self‐field MPD thrusters

In order to propel a manned space craft from earth orbit to Mars various propulsion systems are considered. Propulsion systems which are discussed in connection with manned Mars flight are: chemical rockets, nuclear thermal rockets, ion engines, thermal arc jet thrusters, magnetoplasmadynamic (MPD) thrusters. High power thermal arc jet thrusters and continuously running self-field MPD thrusters offer the capability of being operated at different power levels. This results in variable specific impulse that mainly depends on propellant mass flow rate and electrical power. For each thruster design the attainable specific impulse is limited by a maximum arc current, which can be supplied to this thruster. Therefore, this paper deals with a new thermal plasma thruster which uses two different mechanisms to couple energy into the propellant. ATTILA [8] combines the advantages of thermal arc jets and self-field MPD thrusters, especially their relatively simple setup, and the reliability of a radio frequency power supply, which is usually used for surface hardening. This paper describes the setup of ATTILA and shows its utility as propulsion system. Additionally, numerical calculations to optimize future thruster configurations will be presented.

ATTILA- Adjustable Throttle Inductively Afterburning Arc Jet

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