# Micro-Nozzle Flow ### Abstract --- The accurate numerical prediction of gas flows within micro-nozzles can help evaluate the performance and enable the design of optimal configurations for micro-propulsion systems. Viscous effects within the large boundary layers can have a strong impact on the nozzle performance. Furthermore, the variation in collision length scales from continuum to rarefied preclude the use of continuum-based computational fluid dynamics. ### Introduction --- Predicting the performance characteristics and designing optimal configurations of micro-propulsion systems is vital for the growth in the use of smaller, more cost-effective satellites, such as CubeSats. The CubeSat architecture enables many organizations, including Universities, to have access to low Earth orbit (LEO) without a prohibitive expense. The basic geometric unit is \\(10.\times10.\times10.\\) cm, which results in significant geometric constraints for all elements of the satellite, including the propulsion system. These geometric constraints along with the minimization of the impulse bit size, or change in momentum per pulse, results in very small nozzle requirements for chemical and/or electric propulsion systems, such as resistojets. The small geometric sizes result in low Reynolds number flow, where viscous effects can become significant. Furthermore, the variation in collision length scales between high density chamber and low density exit plane results in multi-scale flows that range from near-continuum (upstream of the throat) to fully rarefied near the nozzle exit plane. In this rarefied regime, there are an insufficient number of collisions to maintain a near-equilibrium velocity distribution function throughout the flow field. Furthermore, energy may be transferred between the translational and internal modes of the gas and the gas composition may change. All of these effects occur within a transient flow due to the heating of the surfaces or cycling of the thruster. This work explores leveraging a high-performance [DSMC library](http://sparta.sandia.gov) to model the unsteady nature of the evolution of the flow from startup, for example from a valve being actuated, to steady-state. These simulation capabilities could enable future work to optimize the nozzle geometry for pulsed thruster configurations, which may be necessary to reduce the thrust bit size for high precision maneuvers. First, the work outlines the [simulation method](#simulation-definition "Simulation Definition"), [geometric configuration](#geometric-configuration), and boundary conditions. Then, we describe some aspects of the steady-state flow. Next, a detailed description of the unsteady buildup of gas in the thruster is examined. Finally, the we conclude with a summary and possible future direction. ### Simulation Definition --- The DSMC library used within this effort is the [SPARTA library](http://sparta.sandia.gov), which is under active development at Sandia National Laboratories. The library is optimized to run efficiently on massive parallel computing architectures. SPARTA overlays a hierarchical Cartesian grid across the simulation domain to track particles, perform collisions, and sample requested statistical quantities. Objects are embedded in the simulation domain using line segments (in 2D and axi-symmetric) or triangulated surfaces (in 3D), which cut through the grid cells. The SPARTA library leverages a scripting interface for its simulation paradigm that results in a highly flexible architecture to simulate, sample, and analyze general transient flows. This is in contrast to most [DSMC implementations](https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo), where the source code must be updated and the software recompiled to define custom transient simulation steps that can vary throughout the simulation. #### Geometric Configuration Following [recent efforts](http://dx.doi.org/10.2514/6.2015-3923), the nozzle body is defined by a third order polynomial, given by: $$ R = az^3 + bz^2 + cz + d $$ where \\(R\\) is the nozzle radius and \\(z\\) is the axial location. The coefficients \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are defined by: $$ a = \frac{2}{L_e^3}\left[r_{tc}-r_e\right]+\frac{1}{L_e^2}\left[r_{tc}'+r_e'\right] $$ $$ b = \frac{3}{L_e^2}\left[r_e-r_{tc}\right]+\frac{1}{L_e}\left[r_e'+2r_{tc}'\right] $$ $$ c = r_{tc}' $$ $$ d = r_{tc} $$ where \\(r_{tc}\\) and \\(r_e\\) are the radius at the starting and ending axial location of the contoured region, respectively, \\(r_{tc}'\\) and \\(r_e'\\) are the slope at the starting and ending axial location of the contoured region, respectively, and \\(L_e\\) is the length of the contoured section. The diverging section is defined by a constant slope, \\(r_d'\\). The throat region is defined by a circular contour based on the radius of curvature \\(r_c\\) and the throat radius, \\(r^*\\). With the definition of the slope on each side of the circular throat, the length of the throat is determined. In this work, the nozzle is defined with the parameters listed in [Table 1]{#table-1}, which corresponds to the geometric definition of a nozzle that is described in Holman and Osborn \cite{Holman-2015}. Because the slope is constant, the coefficients \\(a\\) and \\(b\\) are zero and \\(L_e\\) can be found based on the slope and expansion ratio, \\(\frac{r_e}{r_{tc}}\\). [Figure 1](#Nozzle-Contour) shows a plot of the nozzle contour and the domain of the DSMC simulation. The area ratio between the throat and exit plane is \\(103.9\\) with an overall length of \\(3.75\times10^{-3}\\) m. ![Nozzle-Contour](../static/homepage/examples/micronozzle_flows/img/nozzle_contour.png) Figure 1. Schematic of the micro-nozzle geometry and simulation domain. This effort leverages the axi-symmetric formulation within the ARISTOTLE DSMC library. [Figure 1](#Nozzle-Contour) shows a schematic of the micro-nozzle geometry and simulation domain. The axis of rotation is located at \\(r = 0\\). The \\(z = 0.0035\\) m and \\(r = 0.0009\\) m are vacuum boundary conditions. The \\(z = -0.0005\\) m reference line is defined as an inflow boundary below the chamber radius. At this boundary, \\(\mathrm{N_2}\\) is applied at constant number density and temperature of \\(2.2\times10^{24}\\) \\(\mathrm{m^{-3}}\\) and \\(300.\\) K, respectively. This results in a chamber pressure of approximately \\(9.1\\) kPa. The simulation is initialized as a vacuum within the nozzle. For numerical accuracy, the DSMC method requires that the cell Knudsen number, which is the ratio of the mean free path to the characteristic cell length, be on the order or bigger than unity. ARISTOTLE's DSMC library automatically generated a grid that meets this cell-size restriction, based on the steady-state result. The final grid results in \\(1.44\\) million cells and \\(110\\) million simulated particles. A constant time-step of \\(2.\times10^{-10}\\) s was used throughout the simulation with is less than the chamber mean collision time (minimum) of \\(1.31\times10^{-9}\\) s. [Figure 2](#Nozzle-Steady-State) shows the variation in number density (top) and speed (bottom) for flow of \\(\mathrm{N_2}\\) through the nozzle at steady state. There is nearly a three order of magnitude variation in the number density from the chamber to the nozzle exit plane. Furthermore, the variation in speed shows that the viscous boundary layer is quite thick and occupies about \\(75\%\\) of the area of the exit plane. These strong viscous effects are also seen in [Figure 3](#Nozzle-Steady-State-Temperature), which shows the variation in the translational temperature at steady-state. Beyond the lip of the nozzle exit, there exists a strong expansion, which results in very few molecular collisions. This results in a high translational temperature due to the bi-modal nature of the velocity distribution function separated by gas molecules undergoing free expansion in a vacuum and particles that have collided and thermally accommodated to the diffuse surface. ![Nozzle-Steady-State](../static/homepage/examples/micronozzle_flows/img/ndens_speed_0001050000_shaved.png) Figure 2. Variation in the number density (top) and speed (bottom) for the nozzle flow at steady-state. ![Nozzle-Steady-State-Temperature](../static/homepage/examples/micronozzle_flows/img/temperature_0001050000_shaved.png) Figure 3. Variation in the translational temperature for the nozzle flow at steady-state.