Performance Impacts of Metal Additive Manufacturing of Very Small Nozzles

An analytic model was designed to evaluate whether traditional viscous loss theory predicts the performance of low-thrust nozzles and provides a standard for comparison to the experimental results of this research (Tommila and Hartsfield 2017). The analytic model described in this section is based on the equations of generalized quasi-one-dimensional adiabatic compressible flow with viscous losses as presented by Oosthuizen and Carscallen (1997). The goal of this model is to calculate the thrust coefficient of a nozzle at targeted throat Reynolds numbers and assumed uniform surface roughness. To do so, the local Mach number, pressure, and temperature must be known at various points in the thruster geometry. Viscous losses are accounted for in the calculation of the local Mach number as shown in the ordinary differential equation (ODE) presented in Eq. (1), and numerically solved using Runge-Kutta methods in MATLAB to solve for the square of the Mach number explicitly (Oosthuizen and Carscallen 1997; Shampine and Reichelt 1997)

Viscous effects are captured in Eq. (1) via the Fanning friction factor in the trailing term of the equation. This friction factor is a nondimensional representation of the local shear stress at the wall of the nozzle (Spisz et al. 1965). The equations used to calculate this friction factor depend on the laminar or turbulent nature of the flow. This research adopts the convention of Oosthuizen and Carscallen in considering Reynolds numbers below 2,000 to be fully laminar and those above 2,000 to be fully turbulent. Although the transition region of the flow is certainly more ambiguously defined, extending from 2,000 to 4,000 or 10,000, this approximation is sufficient for the purposes of this research. For laminar flows, the Fanning friction factor is independent of the flow surface roughness and is calculated per Eq. (2) (Oosthuizen and Carscallen 1997)

However, for turbulent flows, the friction factor incorporates the surface roughness via the absolute, or sand grain, roughness of the flow surface (Moody 1944; Oosthuizen and Carscallen 1997). Because this research seeks to distinguish between the performances of machined and LPBF nozzles, the analytic model was evaluated for both surface roughness conditions. Previous research estimated the absolute roughness of the LPBF nozzles from the manufacturer’s specification and assumed that the machined nozzles had the absolute roughness of drawn steel tubing (Tommila and Hartsfield 2017).

The analytic model has been refined to incorporate the average surface roughness measured using a laser scanning microscope (LSM). This surface roughness was then converted to an absolute roughness using an algorithm developed by Adams et al. (2012). Once the absolute surface roughness ($\epsilon$) is known, the Fanning friction factor for turbulent flows is calculated via the methods of Oosthuizen and Carscallen (1997) as shown in Eq. (3)

To reach a closed-form solution of Eq. (1) at all axial points along the model nozzle, the local temperature and pressure must be calculated simultaneously within the solver loop via Eqs. (4) and (5). The chamber conditions in these equations are provided as inputs to the model and are calculated from isentropic theory to target the desired Reynolds number at the throat of the nozzle

Fig. 1 is a graph of the Mach number as a function of axial distance along the model thruster for a typical run of the analytic model. The discontinuity in Eq. (1) at $M=1$ is avoided by stopping the code just prior to the discontinuity and stepping across to a Mach number slightly greater than 1 before continuing the loop. As demonstrated in Fig. 1, a choked flow condition is assumed; the Mach number reaches 1 at the throat of the nozzle. This was calculated for ${\mathrm{Re}}_D=\mathrm{1,000}$ helium flow with a 0.38-mm throat. Although, in the presence of viscous losses, the sonic point will realistically occur in the expansion cone of the nozzle, it is assumed that this approximation is sufficient for the purposes of developing an analytic curve of the accuracy desired for this research (Oosthuizen and Carscallen 1997).

After the solution of Eqs. (1), (4), and (5) at all axial points in the nozzle using the ODE suite in MATLAB version 2016b (Shampine and Reichelt 1997), the thrust coefficient may be calculated. First, the thrust is calculated using the exit pressure, temperature, and Mach number. Assuming vacuum ambient conditions, the thrust equation as presented by Sutton and Biblarz (2010) may be simplified as shown in Eq. (6). After solving for the stagnation pressure in the thruster chamber, the thrust coefficient is calculated per Eq. (7)

Nozzles were manufactured using both EDM and LPBF processes to understand the associated geometries and roughness profiles. Experimental nozzles were designed for testing across a variety of relative throat roughness values and expansion ratios. To vary absolute roughness, nozzles were manufactured using two methods: traditional machining practices, including EDM for the inner surfaces, and additive manufacturing. Fig. 2(a) is the inner surface of the machined nozzle, visually showing a smooth wall, but with an oversized throat due to the manufacturing constraints. Fig. 2(b) shows one of the LPBF nozzles produced for this research, which has a surface with more protrusions; however, it has a more accurate throat diameter. A nominal throat diameter of 0.38 mm (0.015 in.) was targeted for the machined nozzles and two nominal throat diameters, 0.38 mm (0.015 in.) and 0.76 mm (0.030 in.), were targeted for the LPBF nozzles. The manufacturing tolerances had significant variation from these nominal conditions for the machined nozzles, which exceeded the desired throat diameter by approximately 0.15 mm (0.006 in.) or 40%. This highlights the difficulty of consistently producing parts in this size range using traditional manufacturing techniques.

In addition to variation in throat diameter and surface roughness, nozzles were manufactured over a range of expansion ratios. Conventionally machined nozzles were manufactured at measured expansion ratios of 9 and 16. The designed expansion ratios were higher, but the increased throat diameter resulted in a lower than designed expansion ratio. The methodology here, comparing thrust coefficient, minimizes the impact of the throat diameter as long as the calculations include the actual throat diameter, and are compared to theoretical values for the measured expansion ratio. LPBF nozzles were manufactured at a range of measured expansion ratios between 3 and 48. The LPBF nozzles were manufactured by an outside commercial vendor, using Haynes 282 (Haynes International, Kokomo, Indiana) alloy in EOS M290 printers (Krailling, Germany). While many parameters, such as laser power, scan speed, scan strategy, powder sieving, and reuse, can affect surface quality, as shown in Shelton et al. (2020) for cylindrical slots of similar size and printed in a similar alloy, Inconel 718, the specific parameters used by the vendor are considered proprietary information. However, the achieved surface roughness measured in this research, 7.3 μm (Table 1), is near the middle of the range reported by Shelton et al. (2020) for Inconel using all new powder. Haynes 282 has a similar composition to Inconel 718, with more nickel and cobalt and correspondingly lower fractions of some other components. Fig. 3 is a cross-sectional view of a converging-diverging micronozzle produced with LBPF for this research.

The goal of this research was to evaluate the baseline, as manufactured, performance of the additively manufactured nozzles. Many methods exist to smooth the surface of the printed parts in postprocessing, but the first question to answer was if that was necessary. At the scales evaluated in this research and the relatively low pressures and Reynolds numbers, the initial thought, based on historic data, was that the roughness might have negligible impact on performance and the postprocessing might be wasted effort and expense. As shown in the results of this research, there is ample reason to pursue postprocessing at this scale, but in much larger, higher flow nozzles, that effort might not be justified because the relative size of the protrusions into the flow would be much lower.

Previously published work laid out the analytic basis and experimental setup for this research but stopped short of collecting experimental data (Tommila and Hartsfield 2017). Ensuing research has utilized that apparatus to directly measure the performance of a variety of converging-diverging low-thrust nozzles at numerous gas flow conditions. Heated carbon dioxide, nitrogen, or helium flow through the nozzle that is mounted to an inverted pendulum thrust stand in a near-vacuum environment [between 0.13 and 13 Pa (1–100 mTorr), including pressure rise due to gas flow]. Thrust and stagnation pressure were measured directly using a load cell and pressure transducer. The data were collected using LabView, allowing for calculation of the thrust coefficient as presented in Eq. (7). Fig. 4 is a schematic of the experimental setup.

The performance characteristics of each manufactured nozzle were to be measured at a variety of throat Reynolds numbers. The desired throat Reynolds number was achieved by controlling both the temperature of the heater block and the volumetric flow rate of the gas. Combined, these two control mechanisms determine the viscosity, density, and mass flow rate of the gas. First, the vacuum chamber was pumped down [$<11\mathrm{Pa}$ (80 mTorr)] and the heater block was preheated to between 530 and 616 K. Once a steady temperature was achieved, the gas flowed through the mass flow controller into the vacuum chamber, where it passed through the heater block. The gas then passed through the bulkhead of the thrust stand, followed by expansion through the nozzle.

For each test, the nozzle thrust, pressure in the nozzle chamber, and heater block temperature were measured. Measurement of the thrust and pressure resulted in direct calculation of the thrust coefficient via Eq. (4). Monitoring the heater block temperature allowed for verification that the approximate desired Reynolds number was achieved based on flow conditions. Because ambient pressure in the vacuum chamber increased with the introduction of gas flow, the pressure inside the vacuum chamber was also measured and used to correct the output thrust to an equivalent vacuum thrust.

The surface roughness of the machined (316 stainless steel) and LPBF (Haynes 282) nozzles was measured to provide the most accurate estimation of absolute roughness for the analytic model. The Zeiss (Oberkochen, Germany) LSM probed the interior surfaces of two LPBF nozzles and one machined nozzle, measuring both an average surface roughness, $S_A$, and maximum roughness height, $S_Z$. The absolute roughness, $\epsilon$, was then calculated from the average surface roughness using Eq. (8). This equation is the estimation developed by Adams et al. (2012) but adapted for area average surface roughness, $S_A$, rather than line average, $R_A$, to provide a larger geometric statistic of the roughness value over the entire surface. The roughness values for both nozzle types are listed in Table 1 (Tommila 2017). Despite the clear difference in roughness distribution and magnitude evident by the microscope images, Table 1 shows that the average surface roughness of the measured LPBF nozzles is only 14% higher than that of the EDM nozzle

The roughness has been further understood by measurements of 0.4-mm-diameter grooves in printed Inconel 718 (Special Metals, Moore, Oklahoma), which exhibited similar average roughness when measured on an LSM (Shelton et al. 2020). However, there is a 35% difference in the maximum roughness heights between the two nozzle varieties. Although the LPBF nozzles feature large surface protrusions as seen in Fig. 5(a), these surface features are not regularly distributed over the surface of the nozzle and therefore do not contribute appreciably to the average roughness of the nozzle. Because of the low density of these protrusions, this research refers to them as anomalous protrusions.

There is a difference in the distribution of the anomalous protrusions upstream and downstream of the throat as shown in Figs. 5(a and b). Fig. 5(a) is a topography map rendered by the LSM for the additively manufactured nozzle’s cross section, where the various anomalous protrusions are visible in both the converging and diverging sections. Fig. 5(b) shows the distribution by size, measured in 320-μm-square sample regions both upstream of the throat and downstream. Upstream generally has fewer, and smaller, protrusions overall per area, and downstream has larger, more numerous, protrusions. It is thought that this is caused by the orientation of the parts in the printing process. In this case, all of the nozzles were printed with the downstream side facing downward and the upstream part facing upward. Because the expansion angle was only 15° while the contraction angle was 45°, it was initially thought that this orientation would minimize surface roughness overall because overhanging surfaces are, in general, rougher than upward-facing surfaces. In this case, the generally lower density and smaller protrusions on the upward-facing surface would have limited impact on the flow, being in the lower-speed portion of the nozzle and more within the low-speed boundary layer. The higher density and larger protrusions on the downward-facing, diverging side can significantly impact the fluid flow in the nominally supersonic expansion region. In these experiments, the propellant flow should have had little to no erosion or thermal impact on the surfaces because the highest gas temperatures were 616 K and lower and the service temperature of Haynes 282 is in excess of 1,000 K (Haynes International 2020).

To provide a basis for comparison to the experimental results, the analytic model was used to produce theoretical curves of the performance for both the additively manufactured and machined nozzles as a function of Reynolds number and expansion ratio in a style similar to Spisz et al. Fig. 6 shows the results of the analytic model for nitrogen as the propellant gas with a nozzle throat diameter of 0.56 mm (0.022 in.) and expansion angle of 15° (Tommila 2017). Fig. 6 shows that, due to the similarity of the absolute surface roughness of the nozzles, the difference in performance between the two nozzle types is nearly indistinguishable when evaluated using the equations of quasi-one-dimensional compressible flow with viscous losses.

This behavior may be more clearly represented by plotting the Darcy friction factor as a function of Reynolds number in accordance with the methods of Moody (1944) and Tommila (2017). The Darcy friction factor is related to the Fanning friction factor, which incorporates the viscous loss term in the analytic model, by a factor of four. Therefore, the similarity between the friction factors of the EDM and LPBF nozzles shown in Fig. 7 represents the same behavior shown in Fig. 6. The Darcy friction factors, and therefore the viscous losses, for the two nozzles are nearly indistinguishable for turbulent Reynolds numbers investigated in this research.

An alternative representation of the performance predictions produced by the analytic model is as a loss in thrust coefficient when compared to isentropic theory as adopted from Spisz et al. Fig. 8 is a plot of the thrust coefficient loss as a function of expansion ratio and Reynolds number for the same conditions shown in Fig. 6. Because the curves of the EDM and LPBF nozzles are nearly indistinguishable, only the curves for the LPBF nozzle are represented in Fig. 8. The performance shown in Fig. 8 investigates the expected Reynolds number dependence. For low expansion ratios, it is expected that laminar throat Reynolds numbers will outperform turbulent Reynolds numbers in terms of thrust coefficient. Additionally, in the range of turbulent Reynolds numbers investigated, theoretical performance is nearly independent of Reynolds number for nozzles of this relative roughness. Meanwhile, significant variations in performance are expected across all expansion ratios in the laminar regime. These predicted results were produced to simulate all experimental flow conditions and are included in the experimental plots to determine efficacy in predicting measured performance. In all cases, only the analytic curves for estimated LBPF roughness are included in the plots. This provides clarity in the plots without reduction in fidelity given the similarity in expected performance between the two nozzle varieties as predicted by the analytic theory.

The first category of flow scenarios investigated experimentally was for throat Reynolds number in the laminar flow regime. Helium was used as the propellant gas for test cases targeting throat Reynolds numbers of 500 and 1,000. Fig. 9 is a plot of the experimentally measured thrust coefficient for test cases targeting a throat Reynolds number of 500. All nozzles for this test case were manufactured using LPBF to a nominal throat diameter of 0.38 mm (0.015 in.). The analytic model overpredicts the performance of the LPBF nozzles for this laminar flow case. The two data points with mean thrust coefficients exceeding 2.0 represent values that exceed the theoretical maximum thrust coefficient in accordance with isentropic theory. The raw data used to calculate the thrust coefficient in those cases did not exhibit well-defined steady-state behavior and calculations of thrust coefficient because the raw data are likely invalid.

Excluding the two data points showing a $C_F$ of about 2.0, the experimentally calculated loss in thrust coefficient was 2.5 times higher than that predicted by the analytic model for test cases targeting a throat Reynolds number of 500. There was additional potential performance loss due to flow separation in the expansion cone of the nozzle. The mass flow rates required to achieve the desired throat Reynolds number with heated helium resulted in vacuum chamber pressures increasing rapidly. The approximate pressure at the exit of the nozzle calculated for these cases dropped below 40% of the measured ambient pressure for all test cases shown in Fig. 9. Per Sutton and Biblarz (2010), this increase in ambient pressure results in the likelihood of flow separation in the nozzle. This separation would introduce a loss in thrust coefficient unpredicted by the viscous loss theory incorporated in the analytic model. Experimental data for test cases at a Reynolds number of 1,000 exhibited similar behavior. Therefore, it is necessary to examine data taken under conditions unlikely to cause flow separation to assess the validity of the analytic model in predicting losses in thrust coefficient.

Tests cases with higher mass flow targeting transitional and fully turbulent Reynolds numbers of 2,500, 5,000, and 10,000 provided high enough exit pressures to minimize risk of flow separation in the nozzle. Carbon dioxide and nitrogen were used for these test cases. Both machined and LPBF nozzles were tested, including a larger 0.762 mm (0.030 in.) nominal throat diameter variation of the LPBF nozzles.

The first test case targeted a throat Reynolds number of 2,500 for the machined nozzles and small throat variant of the LPBF nozzles. The average throat diameter of the machined nozzles in this test was 0.58 mm (0.023 in.) and the average throat diameter of the LPBF nozzles was 0.46 mm (0.018 in.). The experimentally measured thrust coefficients when compared to analytic viscous flow theory for this case are shown in Fig. 10(a). The analytic model lines for carbon dioxide and nitrogen are included. The thrust performance clearly shows a distinction between the performance of the machined and LPBF nozzles. While the analytic model does not exactly predict either case, experimental uncertainty places the performance of the machined nozzles close to the theoretical prediction line. Meanwhile, the LPBF nozzles experience significantly higher loss in performance than predicted by the analytic model. While Fig. 10 shows that the analytic model predicts carbon dioxide will outperform nitrogen under the same test conditions, this behavior was not always evident in the experimental data. In fact, for both machined nozzle scenarios for ${\mathrm{Re}}_T=\mathrm{5,000}$, nitrogen outperforms carbon dioxide. Figs. 10(b and c) also show the difference in performance between machined and LPBF nozzles at similar expansion ratios and nominal throat Reynolds numbers. In addition to reiterating the gap in performance between machined and LPBF nozzles, the data also display the inconsistent relationship between gas type and nozzle performance.

Because an increase in hydraulic diameter drives a decrease in viscous losses in accordance with the equations of compressible flow through a converging-diverging nozzle, it is expected that the LPBF nozzles with a nominal throat diameter of 0.76 mm (0.030 in.) will outperform the small throat configuration nozzles that were previously analyzed. Fig. 11 plots the experimental thrust coefficient of both configurations and shows that there is not a significant difference in performance between the two groupings, with an average around 1.1.

Viscous loss theory, and the analytic model incorporating this theory, suggests a strong correlation between performance and Reynolds number when comparing laminar flow to turbulent flow. However, experimental results show very little correlation between the loss in performance of nozzles operating at any Reynolds number between 500 and 10,000 as shown in Fig. 12. This shows the experimental thrust coefficient for all small throat diameter configuration LPBF nozzles tested. While supporting the relative independence of performance on Reynolds number in the turbulent cases studied in this research, these results do not demonstrate the expected distinction between laminar and turbulent flow cases.

Fig. 12 also suggests that, while viscous loss theory predicts a strong correlation between nozzle expansion ratio and performance, this behavior is not present in the experimental data. At all Reynolds numbers, it is difficult to determine a distinct relationship between expansion ratio and loss in thrust coefficient. The absence of a distinguishable correlation between nozzle performance and both expansion ratio and Reynolds number suggests that the dynamics of the analytic model are insufficient for prediction of the thrust coefficient of these nozzles. Comparison between the performance of LPBF and machined nozzles widens the gap between analytic and experimental predictions.

One of the primary objectives of this research was evaluating differences in performance between machined and LPBF nozzles. Because of the similarity between the average surface roughness values of the two nozzle types, the analytic model predicts a nearly indistinguishable difference in performance at identical flow conditions. However, experimental results seen in Fig. 13 exhibit a clear distinction in performance between machined and LPBF nozzles at similar flow conditions and geometries. In fact, LPBF nozzles average a 39% loss in thrust coefficient, while the machined nozzles average a 15% loss (Tommila 2017). Fig. 13 plots the average loss in thrust coefficient of the two nozzle types for all targeted Reynolds numbers. This distinction in performance once again suggests that the dynamics of the analytic model do not account for the complexity of the dynamics of the flow. One key feature that is likely present, for which there is no provision in the analytic model, is the presence of shock waves.

The equations of quasi-one-dimensional adiabatic compressible flow with viscous losses, as incorporated in the analytic model, define the dependence of nozzle performance on Reynolds number, propellant gas, and expansion ratio. However, this behavior was not observed in the experimental data. Additionally, while the analytic model did not predict a significant difference in performance between the nozzle varieties, experimental results showed much lower performance in the LPBF nozzles when compared to the machined nozzles. These results suggest that the dynamics of the flow may not be predicted by traditional viscous loss theory as incorporated in this research.

Research performed by Krishnamurty and Shyy (1997) offers a plausible explanation for additional loss terms that may introduce higher magnitude and largely unpredictable performance losses in nozzles of this size. They suggest that in small nozzles with relatively high magnitude, nonuniformly distributed surface roughness features, like those of the LPBF nozzles, the use of average surface roughness in determining viscous effects begins to break down. Additionally, they suggest that viscous loss theory is insufficient in fully quantifying the losses present in supersonic flow through nozzles of this magnitude. Instead, it is likely that large protrusions into the flow in the supersonic, diverging section of the nozzle may produce shock waves within the nozzle. Because of the magnitude of the hydraulic diameter of these nozzles, it then becomes likely that these shock waves interact with each other, resulting in the formation of normal shocks near the centerline, thus further reducing total pressure and thrust. Furthermore, it is possible that vortices form at the tips of the protrusions and form an additional drag term unpredicted by the theory used to produce the analytic model. The shock wave reflection theory presented by Krishnamurty and Shyy would explain the difference in performance between the two nozzle manufacturing methods. The anomalous protrusions likely forming these shock waves were only observed in the LSM images of the LPBF nozzles. Therefore, it is not expected that this behavior would often occur in the machined nozzles due to their smoother surface, with very few, if any, large protrusions above the mean surface. The higher performance and closer adherence to the analytic model observed in the machined nozzles suggests that significant shock waves were not forming.

The theory of shock wave reflection and losses in the performance was evaluated using computational fluid dynamics (CFD). To support the theory, relatively large protrusions were implemented in the CFD. Investigation was performed using a throat diameter of 0.76 mm and an area ratio of 50, similar to the nozzles tested experimentally. The purpose of the study was to better understand what kinds of flow features within the nozzle are causing the overall loss in stagnation pressure at the exit plane. It was believed that the losses are a result of nonisentropic processes caused by the large defects left over as a result of 3D printing. Two geometries were simulated; the first emulated a machined nozzle with no surface defects, and the second emulated a 3D-printed nozzle with a single surface defect, consisting of a 100-μm-diameter hemispherical bump located one throat diameter downstream of the throat of the nozzle. These simulations were conducted using US3D, a finite-volume Navier-Stokes solver (Minneapolis) designed for compressible flows subject to rapid expansion. For further details on US3D, the reader is referred to the work by Nompelis et al. (2006). The simulations used fourth-order accurate kinetic energy consistent flux vector splitting of Subareddy and Candler (2009) employing the limiter by Ducros et al. (2000). Curvilinear structured meshes were used to mitigate the excessive numerical dissipation caused when attempting to resolve shocks on tetrahedral meshes. Similar to what was used in the experimental trials, the test gas was modeled as a calorically perfect nonreacting nitrogen gas. The viscosity of the gas was determined using the nitrogen curve fits by Blottner et al. (1971). In order to more accurately simulate the experimental setup, both meshes included a plenum leading up to the throat of the nozzle, the nozzle itself, and a dump tank at the exit of the nozzle (Fig. 14). This computational domain allows the boundary layer to be completely developed inside the plenum and throat before the flow enters the expanding section of the nozzle. Additionally, the dump tank simulates the low-pressure environment of the vacuum chamber into which the nozzles would exhaust.

A grid convergence study was done on a high Reynolds number case of ${\mathrm{Re}}_T=\mathrm{10,000}$ with the defect near the throat (similar to that shown in Fig. 5). This particular case was chosen to avoid flow separation in the nozzle while still allowing the development of large wakes downstream of the defect. Throughout the grid convergence study, mesh refinement was restricted to the nozzle itself, not to the plenum or dump tank. Additionally, keeping larger element sizes in the plenum and dump tank sections of the mesh aided in dissipating any nonphysical acoustic reflections. As such, the element counts referenced only correspond to the elements within the nozzle section of the mesh (Fig. 15). These mesh sizes correspond to 6.6 million, 10.9 million, and 16.8 million elements with a corresponding mean element spacing of 18, 15, and 13 μm, respectively. The minimum element sizes near the throat and near the walls are considerably smaller than this average value.

The parameters considered in determining grid convergence were the stagnation pressure and streamwise momentum integrated over the entire cross-sectional area of the nozzle at three discrete locations downstream of the throat. These cross sections were located at the throat ($x/L=0.00$), and two defect diameters downstream of the defect ($x/L=0.15$) and at the exit plane of the nozzle ($x/L=1.00$), where $L$ is the total length of the nozzle and $x$ is the cross-section plane along the nozzle. Total thrust at the exit plane was also considered in determining grid convergence. The results of this study are given in Table 2, and are shown on log-log scale in Fig. 16. There was a significant difference in the solution between the coarsest grid and the finest grid, but there was very little difference in the solution between the medium-sized grid and the finest grid. The solution change of the measured thrust shows the expected fourth-order convergence of the numerical method. The solution change in the integrated total pressure shows more rapid convergence at approximately the sixth order. This error metric will be sensitive not only to the discretization error of the numerical method, but also to the spatial resolution of the shock itself, so the observed nonlinear response is not unexpected. With an error less than 2% in all considered fields, it was determined that the 10.9 million element (medium) grid was fine enough to provide accurate use in further calculations.

To understand the entropic losses in the 3D-printed nozzles, a baseline mesh without any defects was created with similar element sizing as the mesh used in grid convergence to directly compare against the mesh with the surface defect present. Simulations were run with subsonic outflow from the dump tank with a back pressure of 11 Pa imposed, similar to the experimental condition. Inflow was determined by solving the subsonic characteristic relations with a stagnation temperature of 598 K and stagnation pressure of 220 kPa. Walls were modeled as no-slip at a fixed wall temperature of 600 K to match the experiment. Figs. 17(a and b) depict the density gradient curves of the solution over each mesh. It is immediately clear that adding the defect [Fig. 17(b)] leads to multiple complex flow features within the nozzle that are believed to be the sources of the loss of stagnation pressure and thrust in the 3D-printed nozzle. Fig. 17(b) highlights these features caused by the defect, including shock-wave formations. Each of the highlighted features contributes to the total entropy production within the nozzle. Fig. 17(c) plots the total integrated entropy along equally spaced cross sections in the nozzle normalized to the integrated entropy at the throat. This clearly indicates that the features depicted lead to a greater rate of increase of entropy and therefore adversely affects the total stagnation pressure and thrust generated from the nozzle.

As shown in the experimental additively manufactured nozzles, there are likely multiple defects of varying size throughout the nozzle rather than just a single one at the throat of the nozzle. Images of grooves with similar dimensions in Inconel 718, from Shelton et al. (2020), show similar features on dozens of parts, indicating that these are inherent features with the laser sintering LPBF process. These additional defects would add even more wakes and shocks within the nozzle that would lead to further increases in entropy and losses in stagnation pressure. As the number of defects increases, the likelihood of shock interactions also increases, leading to potentially higher losses. However, the calculations presented here showed an overall thrust of 0.097 N for the $\epsilon =100$, ${\mathrm{Re}}_D=\mathrm{10,000}$, $P_0=220\mathrm{kPa}$, ${\mathrm{N}}_2$ flow through a 0.76-mm-diameter nozzle. This corresponds to a calculated thrust coefficient of 0.97 from Eq. (7), where an isentropic nozzle of this area ratio would have a vacuum thrust coefficient of 1.88. This loss is mostly associated with an approximately 43% loss of stagnation pressure through the nozzle, resulting in an overall 48% loss of thrust. Referring back to the data presented in Figs. 10(c) and 11, the closest experimental analogs, matching Reynolds number and closest in area ratio, in the experimental data were taken with a Reynolds number of about 10,000, but with actual throats of about 0.45-mm-diameter and area ratios of 28.9, 29.5, 55.2, 57.2, and 61.9. These nozzles delivered mean thrust coefficients of 1.09, 1.13, 1.04, 1.00, and 1.23, respectively, over several runs flowing ${\mathrm{N}}_2$ under similar conditions, with about 12% variation in those values between and during runs, demonstrating fairly high variability between parts, a result attributed by the authors to high variability in the presence, size, and location of these large protruding defects. The numerical results fall within the family of the experimental results. The case illustrated here shows at some level a worst-case location for protruding wall defects, being in a supersonic part of the nozzle, but close to the throat, but even then it is close to the experimental results for similar flows (based on Reynolds number and exit area ratio).