0 Simulation of Low-Density Nozzle Plumes in Non-ZeroAmbient Pressures

nozzle plume using the electron-beam fluorescence technique. Rothe's experiment was numerically simulated The direct simulation Monte-Carlo (DSMC) method by Chung et al. [3] using a continuum code based on the was applied to the analysis of low-density nitrogen plumes Navier-Stokes equations and the direct simulation Monteexhausting from a small converging-diverging nozzle into Carlo (DSMC)method of Bird [4]. The simulation results finite ambient pressures. Two cases were considered that were compared with Rothe's density and rotational simulated actual test conditions in a vacuumfacility. The temperaturedata at various locations inside the nozzle and numerical simulations readily captured the complicated at the nozzle exit plane. In this study, no consideration flow structure of the overexpandedplumes adjusting to the was given to simulation of the finite test-facility pressure. finite ambient pressures, including Maeh disks and barrelIn somewhatrelated work,Pitot-pressure measurementsand shaped shocks. The numerical simulations compared well numericalsimulations were made for a low-density nozzle to experimental data of Rothe. and plume flow by Penko et al. [5] and Boyd et al. [6]. Comparisons were made between continuum and DSMC INTRODUCTION results and Pitot pressure measurements at the nozzle exit plane and various locations in the plume. Comparisons of Low-thrust rocket engines which are used for stationcontinuum and DSMC results were also made for the flow keeping, attitude and altitude control on various satellites inside the nozzle. In these works, test-facility pressures and spacecraft have a significant impact on mission were quite low and, therefore, consideration was not given performance such as on-orbit lifetime, payload, and trip to numericalsimulation of the actual ambient pressure. In time. Another important factor affecting mission other work, Campbell [7,8], Nelson and Doo [9], and performance is contaminationof sensitive instrumentsand Zelesnik et al. [10] also analyzed expanding low-density system components that may be exposed to the thruster nozzle flows using the DSMC method and compared their plumes both in the forward and backward flow directions, results with experimentaldata. Hence, understanding of the detailed flow structure in and The rocket engines under consideration typically have around low-thrust rocket nozzles is important not only for thrust levels under 100 mN, are physically small in size, the accurate prediction of thrust and mass flow but also for and have relatively low operating pressures and mass flow. the precise prediction of the plume flow regions. Reynolds numbers are low, on the order 103, and Low-density flow through small nozzles expanding to rarefaction effects are significant. Under these conditions, vacuum has been examined previously in both the flow contains strong nonequilibrium effects, such as experimental and numerical investigations, though, little slip at a wail, from rapid expansion to near vacuum experimental data are available and most data are for gross conditions. The flow transitions from continuum to freecharacteristics of nozzle performance such as thrust and molecular regimes. Conventionalcontinuum gas dynamics discharge coefficients. Data that provides detailed may not be adequate to accurately analyze the flow and an information on internal flow structurewas published by approachbasedon kinetic theorymay be required. Rothe [1,2] in which density and rotational temperature Of the various methods available for analysis of low" distributionswere measured insidea small nozzle andin the density gas flows, the DSMCmethod is most widely used and readily applicable. The DSMC method is a numerical simulation technique for solving the Boltzmann equation * NASAResident ResearchAssociateatLewisResearchCenter. by modeling a real gas flow using a representativeset of "*Professor,MemberAIAA. molecules. Theoretically, the DSMC method can be t Chief,Computational MethodsforSpaceBranch, applied to any flow for which the Boltzmann equation is MemberAIAA. valid but intensive computational requirements generally * AerospaceEngineer,Member IAA. restrict the use to near continuum and rarefied flows. Continuum methods are usually much more efficient than condition. Likewise, the downstream boundary BC is the DSMC method for higher density flows. Thus, in the located far enough from the nozzle exit so that extending it analysis of flows which involve both continuum and further downstream does not result in any significant rarefied regimes, it is reasonable to apply both methods, changes in the macroscopic flow variables in the near The simplest utilization of both methods is to solve the plume. Along the boundary BC, several boundary rarefied flow regime with the DSMC method using conditions are tested including an equilibrium condition boundary conditions for the inflow surface obtained from corresponding to the ambient condition, a vacuum ' the continuum method used to solve the continuum and boundary condition, and an equilibrium condition near-continuum regime, corresponding to a profile extrapolated from the inside of In this paper, the DSMC method is employed for the the plume. The test gas is nitrogen with a stagnation , analysis of low-density nitrogen flows expanding through a temperature of To = 300 K. The flow conditions are listed small converging-diverging nozzle and into finite back in Table 1. In the table, the throat Reynolds number, pressures that simulate Rothe's experimental conditions. Re,t = 2th/_goRt, is based on the viscosity at the stagnation Special attention is given to the effect of the non-zero chamber condition, go. Here the quantity rh is the mass ambient pressure on the flow structure, both in the nozzle flow rate and Rt is the throat radius. The Knudsen number but especially in the plume, to simulate conditions often Kn is based on the throat diameter and the stagnation encountered in ground-based test facilities used to test chamber condition. small thrusters. In contrast to the behavior of low-density plumes expanding into a vacuum, the flow expandinginto a region of finite pressure is often overexpanded witha more DSMC METHOD complicated flow pattern involving Mach disks and shock waves. Under these conditions, the supersonic flow can be The DSMC code used in this study is based on the confined to a narrow core, depending on the applied principles described by Bird [4], together with the variable background pressure, and the sonic line may not intersect hard sphere (VHS) model [11] as a molecular model and with the nozzle lip as it does with a vacuum ambient, in the no time counter (NTC) method [12] as a collision which case the external conditions can influence the sampling technique. The code was developed to internal flow through the thick subsonic region near the investigate various low-density flows of gas mixtures in nozzle wall. These conditions are illustrated in the results arbitrarily shaped flow domains [3,13,14]. Details of the from the numerical simulation and in comparison with code may be found in Ref. 3. The execution speed of the Rothe's experimental data. code for the flow considered in this study, measured by CPU time/particle/timestep, is about 1.3gs on a CRAY PROBLEM STATEMENT Y/MP. The flow domain consists of about 20,000 cells in 41 subregions. At the steady phase of the simulation, the Rothe's experiment [2] was chosen as a reference total number of simulated molecules in the flow domain is problem because of the availability of detailed about 2 million. The flow field is sampled every 5 measurements. Figure 1 illustrates the geometry of the timesteps during 20,000 timesteps after reaching the steady nozzle used in Rothe's experiment and in the numerical phase. The total CPU time required for the computation is simulation. The actual nozzle was made of graphite to about 19hours on the CRAY Y/MP. reduce optical reflections and to minimize back-scattering The VHS exponent, o_,of nitrogen is chosen to be 0.24 and secondary emission of electrons. The subsonic and with the reference molecular diameter of 4.07x10"l°m at supersonic portions of the nozzle are cones having halfthe reference temperature 273K [11]. Chemical reactions angles of 30° and 20°, respectively, with longitudinal radii and the vibrationalmode are assumed to be frozen. For the of curvature at the throat equal to 1/2 of the throat radius, calculation of rotational energy exchange between the The area ratio at the exit based on the throat area is 66. colliding molecules, the Borgnakke-Larsen phenomenoThe shaded region in Fig. 1 indicates the domain of the logical model [15] is employed together with the DSMC simulation. The length of the curved contour temperature-dependent energy exchange probability of downstream of the throat (IH) is about 0.5 mm. The Boyd [16] modified by Chung et al. [3] to be consistent simulation domain extends from the throat to an axial with the experimental data for the rotational relaxation of distance 260 mm from the nozzle exit plane (A.B),a radial nitrogen obtained by various methods and compatible with distance 50 mm from the nozzle lip (GD), and an axial the VHS model. A diffusely adiabatic wall with 10% distance 20 mm from the nozzle exit plane into the thermal accommodation is assumed for the interaction backflow region (DE). The inflow boundary is located at between the gas molecules and the wall [3]. the nozzle throat (OI). The boundary condition at the To assess the effect of backseattering of downstream , inflow boundary is obtained from a solution of the Naviermolecules and reduce the number of simulated molecules, Stokes equations [3]. The radial boundary CE is located the gas is treated as a mixture consisting of two different far enough from the axis so that extending it further nitrogen sources whose origins are the upstream reservoir radially does not result in any significant changes in the and t