CFD Numerical Simulations of Supersonic Flow in a Linear Aerospike Micronozzle.pdf

Numerical Simulations of Supersonic Flow in a Linear Aerospike Micronozzle A. Zili´c ∗ D.L. Hitt ∗ A.A. Alexeenko † ∗ School of Engineering, University of Vermont † School of Aeronautics & Astronautics, Purdue University In this study, we numerically examine thrust performance of the linear aerospike nozzle micro-thruster for various nozzle spike lengths and flow parameters in order to identify optimal geometry(s) and operating conditions. Decomposed hydrogen-peroxide is used as the monop
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  Numerical Simulations of Supersonic Flow in aLinear Aerospike Micronozzle A. Zili´c ∗ D.L. Hitt ∗ A.A. Alexeenko † ∗ School of Engineering, University of Vermont † School of Aeronautics & Astronautics, Purdue University In this study, we numerically examine thrust performance of the linear aerospike nozzlemicro-thruster for various nozzle spike lengths and  fl ow parameters in order to identify optimalgeometry(s) and operating conditions. Decomposed hydrogen-peroxide is used as the monopro-pellant in the studies. Performance is characterized for di ff  erent  fl ow rates (Reynolds numbers)and aerospike lengths, and the impact of micro-scale viscous forces is assessed. It is found that2-D full micro-aerospike e ffi ciencies can exceed axisymmetric micro-nozzle e ffi ciencies by asmuch as 10%; however, severe penalties are found to occur for truncated spikes at low Reynoldsnumbers. I. Introduction In recent years signi fi cant research has been conducted in the area of micropropulsion systems for orbitalmaneuvering of next-generation miniaturized satellites (‘micro-’, ‘nano-’ and ‘pico-sats’) having masses ∼ 10 kg orless. A key component of any chemical-based micropropulsion scheme is the supersonic nozzle used for convertingpressure energy of the combustion gases into thrust. In the aerospace literature it has been documented that the fl ow in supersonic micro-nozzles can be substantially a ff  ected by viscous e ff  ects, thus reducing the performance of the thruster. 1 − 6 For the various linear micro-scale nozzles reported in the literature the Reynolds numbers aretypically quite low: typical values are well below 1 , 000 and some less than 500. Consequently, viscous subsonic‘boundary layers’ form on the expander wall section which can become su ffi ciently thick so as to retard the bulk fl ow and lower the e ffi ciency.The focus of the present study is a computational investigation of the performance of a 2-D micro-scale  linear aerospike   nozzle design. To the best of our knowledge, this represents the  fi rst such report in the aerospaceliterature. For micropropulsion applications, there is little need for the de fi ning pressure-compensation attributeof the aerospike since the ambient conditions are those of either space or near-space. However, one can seek toleverage the fact that a virtual (free) boundary can potentially  mitigate viscous losses   known to occur for internalmicronozzle  fl ows with multiple solid boundaries. This micro-nozzle concept is also worthwhile of investigationowing to its amenability to existing micro-fabrication techniques, a trait not shared by axisymmetric nozzles onthe micro-scale.In this paper, we numerically investigate the thrust production and e ffi ciency of full-length aerospike as well as20% and 40% truncated geometries (‘plug’ nozzles) for a range of Reynolds numbers ( <  10 3 ) based on estimatedmass  fl ow rates for targeted nanosat thrust levels. The working gas is chosen to be a fully-decomposed, 85%hydrogen-peroxide monopropellant. The joint consideration of truncated spikes is based on the prevalence of plug designs in macro-scale nozzles. The latter is often driven by a thrust-to-weight analysis — it has beenexperimentally observed that the majority of thrust for a macro-scale linear aerospike is generated over the  fi rstquarter of the spike. On the micro-scale, the weight savings of a spike truncation is negligible; however, spiketruncation instead carries with it implications for boundary layer growth and possible  fl ow separation. It is foundthat micro-scale aerospikes o ff  er an attractive alternative in performance compared to 3D linear micro-nozzles. 1 of   1 4 American Institute of Aeronautics and Astronautics 37th AIAA Fluid Dynamics Conference and Exhibit25 - 28 June 2007, Miami, FL AIAA 2007-3984 Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.  II. Computational Model A. Numerical Methods for Continuum FlowsH 2 O 2  Monopropellant . In this work, the continuum-based  fl ow analyses are focused on the performance of micronozzle  fl ows featuring decomposed monopropellant hydrogen-peroxide srcinally proposed by Hitt et al. 7 The complete decomposition of the monopropellant fuel is assumed to have occurred upstream of the nozzlewithin the catalytic chamber. The decomposition of the hydrogen peroxide monopropellant proceeds accordingto the one-step reaction2 H  2 O 2 ( l ) → 2 H  2 O ( g ) +  O 2 ( g ) +  heat.  (1)In the numerical simulations, the thermophysical properties of the gas mixture are assumed to be temperature-dependent and are calculated as a mass-weighted average of the mixture components. The inlet temperature isequal to the fully decomposed adiabatic  fl ame temperature of 85% concentration  H  2 O 2  and is set at 886 K   andserves as an inlet condition to the nozzle. The corresponding Reynolds number for the  fl ow is Re  =  ˙ mLµA  (2)where  ˙ m  is the mass  fl ow rate per unit depth,  L  is the characteristic length scale (e.g., the nozzle throat diameter), µ  is the dynamic viscosity of the decomposed monopropellant, and  A  is the cross-sectional area. The value of   ˙ m can be well estimated from quasi-1D theory according to 8 ˙ m  =  p 0 A ∗ √  T  0 s  γ  R µ  2 γ   + 1 ¶ ( γ  +1) / ( γ  − 1) (3)where  A ∗ is the nozzle throat area,  γ   is the ratio of the specifc heats, and  R  is the gas constant. Aerospike Geometry & Computational Domain.  The idealized spike geometry can be generated usingthe approach of Angelino. 9 In short, this inviscid approach combines Prandtl-Meyer expansion theory with thearea-Mach relation for quasi-1D nozzle  fl ow and determines the spike boundary as a particular streamline in the fl ow downstream of the nozzle throat. The resulting spike geometry is shown in Figure 1. The length of thenozzles spike is the only varying geometric parameter; here we considered a full length spike, as well as 20% and40% truncations. A schematic of the entire computational domain is shown in Figure 2. The throat dimensionis identical for every micro-nozzle in this study and is maintained at 90 microns  to match the NASA prototypemicro-nozzle described in Hitt et al. 7 The GAMBIT2.1 grid generation software (Fluent Inc.) was used to de-velop the two-dimensional computational meshes. Depending on the spike length, the grid size contained between180 , 000 and 250 , 000 elements. The meshes have been re fi ned to the point where simulations are independent of further grid re fi nement. Governing Equations.  Continuum modeling is assumed in this study; this can be justi fi ed  a posteriori   byperforming a Knudsen number analysis of the computed  fl ow  fi eld in the regions of interest. We note in passingthat portions of the supersonic plume downstream of the spike will certainly be rare fi ed (non-continuum) andthe model accuracy is degraded; however, these areas are su ffi ciently removed from the spike region such thatthrust prediction is not compromised. The micronozzle  fl ow  fi eld is thus governed by the compressible Navier-Stokes Equations which are solved using a coupled implicit solver for all simulations. The governing conservation 2 of   1 4 American Institute of Aeronautics and Astronautics  equations for mass, momentum, and energy are given by ∂ρ∂  t  + ∇ ·  ( ρ V ) = 0 (4) ∂ ∂  t ( ρ V ) + ∇ ·  ( ρ VV ) =  − ∇  p  + ∇ ·  ( τ  ) (5) ∂ ∂  t ( ρ E  ) + ∇ ·  ( V ( ρ E   +  p )) =  ∇ ·  ( k ∇ T   + ( τ   · V )) (6) E   =  h −  p ρ  +  V 2 2 (7) τ   =  µ µ³ ∇ V + ∇ V T  ´ − 23 ∇ · VI ¶ .  (8)In these equations,  E   is the speci fi c energy, and  p  is the absolute local pressure,  µ  is the  fl uid viscosity,  k  is thethermal conductivity,  T   is the static temperature,  h  is the enthalpy, and  τ   is the viscous stress tensor. Thesystem is closed by the ideal gas law equation of state  p  =  ρ RT   (9)No slip boundary conditions are imposed on the nozzle walls. Subsonic portions of the outlet boundaries areprescribed a constant back-pressure value of   p ∞ =1.0 kPa. This value serves to maintain the Knudsen numberwithin the continuum regime for the aerospike region. For supersonic portions of the domain outlet, the pressureand all other  fl ow quantities are extrapolated from the interior  fl ow via the method of characteristics (Riemanninvariants). A prescribed pressure is imposed at the inlet boundary, which is regarded as a stagnation value.Inlet gas properties are determined at the fully decomposed adiabatic  fl ame temperature of 85% pure hydrogenperoxide, 886  K  . Varying the pressure inlet boundary condition allows performance to be investigated at di ff  erentthroat Reynolds numbers (mass  fl ow rates). In these simulations, the inlet pressure was varied from  p 0  = 25 kPato  p 0  = 250 kPa results in a throat Reynolds number in the range of 60 to 830 and nozzle pressure ratios  p 0 /p ∞ ranging from 25:1 to 250:1.The compressible Navier-Stokes equations were solved using the coupled implicit solver within the FLUENT6.2software package. The  fi rst phase of iterations is done in the  fi rst-order descretization scheme, once the  fi rst-orderconvergence is determined, second order descretization schemes are implemented for the  fi nal solution convergence.The convergence of solution is determined by residuals and  fl ow monitors established at key locations within thedomain. The controlling  fl ow monitors were placed at the locations where high pressure temperature and velocitygradients were identi fi ed. Conservation of mass through the nozzle was also carefully monitored and veri fi ed. B. Rare fi ed Flow Modeling via DSMC To provide an independent comparison to the continuum-based model predictions, rare fi ed  fl ow calculations werealso performed. Aside from simple validation, these calculations were intended to quantify the degree to whichnon-continuum e ff  ects contribute to predictions for thrust output. The direct simulation Monte Carlo (DSMC)code SMILE 10 was used to simulate aerospike nozzle expansion into vacuum. The hydrogen peroxide was modeledusing the variable hard sphere model with the molecular diameter  d  = 4 . 17  ×  10 − 10 m, viscosity-temperatureexponent  α  = 0 . 31 and the Larsen-Borgnakke model for internal energy relaxation with collisional numbers Z  r  = 5 and  Z  v  = 50. The SMILE code uses two-level rectangular cells with automatic grid adaptation. Thebackground collision cell size was 10 µm  with a maximum cell partition level of 10. The in fl ow boundary forthe DSMC calculations was located at the aerospike nozzle exit and the exit velocity, temperature and pressurewere based on the Navier-Stokes solution described above. For the case of vacuum expansion at inlet pressure of 25kPa, the total number of simulated molecules was about two million and the calculations took four hours usingeight AMD Opteron 885 processors. III. Numerical Results Steady-state  fl ows have been computed for full and truncated spikes for Reynolds numbers ranging from 64-830; plots showing Mach contours and streamline patterns appear in Figure 3. Overall, the  fl ow behavior on 3 of   1 4 American Institute of Aeronautics and Astronautics  P inlet 100%Inv. 100%Vis. 40% Inv. 40%Vis. 20% Inv. 20%Vis. DSMC 100%250  31.7 29.7 30.1 24.2 27.2 23.9 28.1 200  26.9 22.7 23.9 21.3 21.7 18.9 - 150  19.8 16.8 18.1 15.5 17.1 13.9 - 100  13.5 12.0 11.8 10.0 10.8 9.0 - 75  10.1 9.0 9.0 7.5 8.2 6.5 - 50  7.6 4.9 6.0 4.5 5.4 4.1 - 25  4.0 2.4 3.0 1.9 2.8 1.8 1.93 Table 1. A summary viscous, quasi-inviscid and DSMC thrust values for the full aerospike and two truncatednozzles (20%, 40%). The pressure inlet values are in kPa and the thrust values are in  µN   per unit depth. Thebackpressure in all continuum cases is 1.0 kPa, and the backpressure for the DSMC case is a virtual vacuum at  ∼ 0 kPa. the micro-scale exhibits many of the same general features reported in the aerospace literature for large-scaleaerospike nozzles. 11 As expected for a  fi xed geometry, the changing Reynolds number alters the free boundaryre fl ection mechanism — and hence the plume characteristics — corresponding to the varying pressure ratios  p 0 /p ∞ .For the truncated spikes one also expects (from inviscid theory) somewhat di ff  erent plume characteristics at a fi xed  Re  since the expansion ratio  A/A ∗ varies with spike length. Viscous e ff  ects are clearly evidenced by the  fl owseparation at the base of the truncated spikes. At low Reynolds numbers, a notable viscous subsonic boundarylayer forms along the surface of the spike and grows in thickness with downstream distance. Similar phenomenahas been observed on the expander walls of supersonic linear micronozzles. For su ffi ciently low Reynolds numbersthe subsonic layer can become su ffi ciently ‘thick’ so as to retard  fl ow in the vicinity of the spike, which is thedownstream of the throat. Shown in Figures 4-5 are the computed subsonic layer thicknesses at three di ff  erentReynolds numbers for the full spike and 20% spike con fi gurations. The signi fi cant extent of the subsonic regionis quite evident at the lowest values of the Reynolds numbers.Thrust production has been calculated for full and truncated (20%, 40%) aerospikes for the range of inletpressures (i.e.,  Re ) and the results are plotted in Figure 6. It is found that there is very little di ff  erence in thethrust production at the lowest inlet pressure levels. Di ff  erences appear at the higher inlet pressures and  fl owrates, with the maximum thrust being yielded by the full spike. The results for the higher  fl ow rates can beunderstood in terms of the e ff  ective expansion ratio ( A/A ∗ ). The relatively poor performance of the full spike atthe lowest  fl ow rates can be attributed to the mergence of signi fi cant viscous losses. Detailed examination of the fl ow  fi eld shows that the end result depends jointly on the viscous subsonic layer size and subsonic  fl ow turningat the ends of the truncated spike.Tabulated values of calculated thrust is compared with estimates obtained from quasi-inviscid simulationsfor di ff  erent nozzle con fi gurations and inlet pressures in Table 1. The degradation in performance arising fromviscous e ff  ects is quite evident in this data. Of particular note is the relatively poor performance of the truncatedaerospikes. On the macro-scale, this is a practical design consideration in which substantial weight savings canbe obtained with modest reductions in thrust production, as evidenced by the inviscid results here. However, onthe micro-scale, the  fl ow separation e ff  ects at the lower Reynolds numbers result in much more severe penalties inperformance — moreover, the savings in weight associated with a full aerospike is virtually negligible for a MEMSdevice. Thus, from a design perspective, it appears that only full (or nearly so) aerospike con fi gurations shouldbe considered. Comparison with Rare fi ed Predictions.  To assess the signi fi cance non-continuum  fl ow e ff  ects, a limitednumber of comparisons of thrust predictions have been made with those obtained by DSMC calculations. Figure7 shows a comparison of the continuum and DSMC density  fi elds in the region of the full spike for a pressureinlet of 25 kPa. In general, the  fl ow behavior is quite similar. The thrust results for the full spike at thetwo extreme inlet pressures of 25 kPa and 250 kPa appear in Table 1. It should be noted when making thiscomparison that the DSMC calculations were performed with a vacuum backpressure condition rather than 1 kPaowing to boundary condition constraints associated with the numerical algorithm. It is seen that at the highest fl ow rate the continuum and DSMC results di ff  er by approximately 5% whereas almost a 25% di ff  erence is found 4 of   1 4 American Institute of Aeronautics and Astronautics

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