Modeling viscous effects on boundary layer of rarefied gas flows inside micronozzles in the slip regime condition

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73rd Conference of the Italian Thermal Machines Engineering Association (ATI 2018), 12-14 73rd Conference of the Italian Thermal Machines September 2018, Engineering Pisa, Italy Association (ATI 2018), 12-14 September 2018, Pisa, Italy

Modeling viscous effects on boundary layer of rarefied gas flows ModelingThe viscous effects on boundary layer of rarefied gas 15th International Symposium on District Heating and Cooling flows inside inside micronozzles micronozzles in in the the slip slip regime regime condition condition a,∗ a Maria Grazia Giorgi , Donato Fontanarosa , Antonio Ficarellaa Assessing theDefeasibility of using the heat demand-outdoor Maria Grazia De Giorgia,∗, Donato Fontanarosaa , Antonio Ficarellaa University of Salento, Dep. of Engineering for Innovation, Via per Monteroni, 73100-Lecce, Italy temperature function for a long-term heat demand forecast University of Salento, Dep. of Engineering for Innovation, district Via per Monteroni, 73100-Lecce, Italy a a

I. Andrića,b,c*, A. Pinaa, P. Ferrãoa, J. Fournierb., B. Lacarrièrec, O. Le Correc Abstract Abstract a IN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal The present work provided abnumerical investigation of the supersonic flow of rarefied gas into a planar micronozzle characterized Recherche & Innovation, Avenue Dreyfous Limay, Francemicronozzle characterized The present work provided a Veolia numerical investigation of the291 supersonic flow of Daniel, rarefied78520 gas into a planar by small depth c and long divergent section. 2D and 3D computational fluid dynamics (CFD) computations were performed using Département Systèmes Énergétiques et Environnement IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, by depthNavier-Stokes and long divergent section. 2D and 3D with computational dynamics (CFD) computations were France performed the small continuum equations in combination partial slipfluid conditions at walls, based on a the establishment of theusing slip −3 −1 the continuum Navier-Stokes equations in combination with partial slip conditions at walls, based on a the establishment of the slip i.e. regime related to a Knudsen number ranging between 1 × 10−3 and 1 × 10−1 . Different partial slip conditions were considered, × 10 . Different partial sliponconditions considered, regime to pure a Knudsen number ranging 1 × 10 the idealrelated case of slip conditions and the between full viscous case and with1Maxwellian slip conditions sidewalls were and planar walls,i.e. as the ideal slip conditions full viscous case withslip Maxwellian on sidewalls and planarcoefficient walls, as well as thecase caseofofpure Maxwellian slip justand on the sidewalls. The Maxwell model wasslip set conditions with a tangential accommodation Abstract well the case to of0.8. Maxwellian slip just onbased sidewalls. The Maxwellof slip set with a tangential accommodation coefficient equalas(TMAC) Comparisons were on the estimation themodel globalwas performance of the micronozzle in terms of thrust equal (TMAC) to 0.8. Comparisons were based on the estimation of the global performance of the micronozzle in terms of thrust force, specific impulse, discharge coefficient and I -efficiency. sp District heating networks are commonly addressed in the literature as one of the most effective solutions for decreasing the force, specificthat impulse, discharge coefficient I sp -efficiency. It resulted when the nozzle depth was and neglected, 3D simulations led tohigh the same solutionwhich obtained meansthrough of 2D compugreenhouse gas emissions from the building sector. These systems require investments are by returned the heat It resulted that when the nozzle depth was neglected, 3D simulations led atolinear the same solution byand means of 2D computations inside the micronozzle. The boundary layer thicknesses experienced growth on theobtained sidewalls, the could viscous losses sales. Due to the changed climate conditions and building renovation policies, heat demand in the future decrease, tations inside the micronozzle. The boundary layer thicknesses experienced a linear growth on theinsidewalls, and the viscous losses produced a reduction of the performance of about the 95%. Significant differences were found the prediction of the jet plume, prolonging the investment return period.of about the 95%. Significant differences were found in the prediction of the jet plume, produced a reduction of the performance which took scope the typical bell-shape form in cases involving of 2Dusing computations, yet 3D simulations estimated a function plume characterized by The main of this paper is to assess the feasibility the heat demand – outdoor temperature for heat demand which took the of typical bell-shape form in cases involving computations, yet 3D estimated a plume characterized by the succession oblique shock waves and expansion fan2D waves. Instead, when thesimulations nozzle depth was considered, 3D simulations forecast. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 the succession of obliquedifferent shock waves and of expansion fan waves. Instead, when the of nozzle depth was considered, 3D simulations underlined a completely behavior the flow because of the establishment the nozzle blockage and a viscous heating. buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district underlined a completely behavior of the of flow because of theand establishment blockage and a the viscous heating. The performance suffereddifferent an intense degradation about the 47%, the analysisof ofthe thenozzle jet plume highlighted formation of renovation scenarios were developed (shallow,ofintermediate, deep). To estimate thethe error, obtained heat demand values were The performance suffered an intense degradation about the 47%, and the analysis of jet plume highlighted the formation of the Mach disk followed by the typical diamond-shaped subsonic recirculation region. compared with results from dynamic heat demand model, previously developed and validated by the authors. the Mach disk followed by thea typical diamond-shaped subsonic recirculation region. The results showed that when only weather change is considered, the margin of error could be acceptable for some applications c(the  2018 TheinAuthors. Authors. Publishedwas by Elsevier Elsevier Ltd.20% for all weather scenarios considered). However, after introducing renovation © 2018 The by Ltd. error annual Published demand than c  2018 The Authors. Published by lower Elsevier Ltd. This is an open access article under the CC BY-NC-ND license This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) scenarios, the access error value increased upCC to BY-NC-ND 59.5% (depending the weather and renovation scenarios combination considered). This is an open article under the license on (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the scientific of the the 73rd 73rd Conference Conference of of the the Italian Italian Thermal ThermalMachines Machines committee The value of slope coefficient increased on average within the range of ofthe 3.8% to 8% perofdecade, thatThermal corresponds to the Selection andAssociation peer-review(ATI under responsibility of the scientific committee of 73rdupConference the Italian Machines Engineering 2018). decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and Engineering Association (ATI 2018). renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the Keywords: micro-power systems; planar micronozzle; viscous effects; boundary layer; gas rarefaction; OpenFOAM micro-power systems; planar micronozzle; effects; boundarythe layer; gas rarefaction; OpenFOAM Keywords: coupled scenarios). The values suggested couldviscous be used to modify function parameters for the scenarios considered, and improve the accuracy of heat demand estimations. © 2017 The Authors. Published by Elsevier Ltd. ∗ Corresponding author. Address: University of Salento, Dept. of Engineering for Innovation, Research Center for Energy and Environment Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and ∗ Corresponding author. Address: Salento, Dept. of Engineering for Innovation, Research Center for Energy and Environment (UNISALENTO-DII-CREA), Via perUniversity Monteroni,ofLECCE I-73100, Italy. Tel: +39 0832297759. Cooling. (UNISALENTO-DII-CREA), Via per Monteroni, LECCE I-73100, Italy. Tel: +39 0832297759. E-mail address: [email protected] E-mail address: [email protected] Keywords: Heat demand; Forecast; Climate change

1876-6102 2017The TheAuthors. Authors.Published PublishedbybyElsevier ElsevierLtd. Ltd. c©2018 1876-6102 1876-6102  © 2018 The TheAuthors. Authors. Published by Elsevier Ltd. c under 1876-6102 2018 Published by Elsevier Ltd. Peer-review responsibility ofthe theCC Scientific Committee of The 15th International Symposium on District Heating and Cooling. This is an open access article under BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection responsibility of the scientific committee of theof73rd of the Italian MachinesMachines Engineering Selection and andpeer-review peer-reviewunder under responsibility of the scientific committee the Conference 73rd Conference of the Thermal Italian Thermal Selection and peer-review of the scientific committee of the 73rd Conference of the Italian Thermal Machines Engineering Association (ATI 2018). under Engineering Association (ATI responsibility 2018). Association (ATI 2018). 10.1016/j.egypro.2018.08.113

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1. Introduction In the last twenty years, the interest in the micro-power systems increased due to the advancements in microfabrication technology of Micro Electro-Mechanical System (MEMS) devices. In the space field, both scientific and industrial communities focused on the development of small spacecrafts for Earth observation and space surveillance. They need for propulsion systems able to ensure an accurate attitude control in order to perform small orbital maneuvers which require small thrust forces from few micronewtowns up to some millinewtons, with stringent constraints of mass, volume and power consumption. Recently, several types of MEMS-based micro-propulsion systems, or microthrusters, have been proposed to be placed on-board small satellites [1]. The micronozzle is a key component for each of them, since the global performance of the entire microthruster is strictly related to its efficiency, which depends on the entity of the viscous effects and the gas rarefaction condition. Due the complexity introduced by the micro-scale, the computational fluid dynamics (CFD) is an essential tool for the characterization of micronozzles and the investigation of the supersonic flow inside them. In this regard, Bayt and Breuer [2] conducted 2D numerical simulations on micronozzles for a cold gas microthruster and concluded that the growth of the subsonic boundary layer reduced the thrust efficiency due to the viscous losses and the reduction of the actual cross section at the nozzle exit. Similarly, Louisos and Hitt [3, 4] performed a deep numerical investigation of the supersonic flow inside a linear micronozzle for monopropellant thrusters. By means of 2D and 3D simulations, they found that an half-angle of the expander equal to 30◦ mitigated viscous losses. Also, they demonstrated the benefit gained in using as large as possible depths. The growth of the boundary layer over the parallel walls produces another negative relevant phenomenon, namely the nozzle blockage, as observed by Handa et al [5] who investigated underexpanded flows into micronozzles with planar geometry. Furthermore, they observed a propagation of pressure waves upstream into the nozzle through the subsonic regions of the boundary layer developed along the inner sidewalls. Other studies focused on the influence of non-conventional expander geometries in reducing of the viscous effects. Cheah and Chin [6] proposed a novel two-depth design and conducted computational fluid dynamics (CFD) simulations which demonstrated that the novel geometry outperformed with respect to the conventional planar configuration. Similarly, Solokov and Chernyshov [7] investigated the effectiveness of the expander of axisymmetric tronco-conical micronozzles at zero ambient pressure with sidewalls partially inclined normal to the axis of symmetry, which led to improved performance owing to the appearance of local recirculation zones. The degree of gas rarefaction also affects the entity of the viscous effects, since it determines the mechanisms of interaction between gas-gas molecules and solid wall-gas molecules. It represents a big concern in numerical modeling, since different approaches exist in relation to the the specific rarefied gas regime. In general, in micronozzles the continuum assumption and the no slip condition at walls are violated and two different regime could establish based on the entity of the Knudsen number: the slip flow regime (0.01 < Kn < 0.1) or the transitional flow regime (0.1 < Kn < 10) can take place. In cases involving the first regime, the Navier-Stokes equations are still valid even though they have to be used in combination with partial slip models at walls [8]. Instead, when the transitional flow regime is verified, gas kinetic schemes such as Direct Simulation Monte Carlo (DSMC) [9] are required, as in [10] and [11]. Liu et al. [12] demonstrated that the continuum method with slip boundary conditions provided a good estimation of the boundary layer inside the nozzle, albeit in the nozzle exit lip region the DSMC methods performed better due to the gas rapid expansion and the enhanced rarefaction effects. Despite the relevance of the scientific works dealing with the analysis of supersonic flows inside micronozzles, a deep understanding of the strong coupling between viscous and gas rarefaction effects is still far from being reached. This reflects on the numerical modeling of such kind of flows. In this context, the present work is aimed to provide useful insight of the viscous behavior of the supersonic gas flow in slip regime into a planar micronozzle designed for a water-propelled vaporizing liquid microthruster characterized by small depth and long expander. For the purpose, CFD simulations were performed in both 2D and 3D configurations using the continuum Navier-Stokes equations in combination with the Maxwell slip model at walls with a tangential momentum accommodation coefficient (TMAC) equal to 0.8. The analysis was extended by simulating the supersonic flow in pure slip conditions, which allowed for the estimation of the global performance of the micronozzle in terms of thrust force, specific impulse, discharge coefficient and I sp -efficiency. They were used in combination with a qualitative description of the flow behavior, in order to highlight differences and peculiarities of numerical solutions resulting from the several investigated cases.

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2. Numerical approach 2.1. Governing equations The CFD investigation of the gas flow through the micronozzle was performed by using the open source CFD toolc Version 3.0.1, based on a Finite Volume formulation. The density-based solver rhoCentralFoam box OpenFOAM was used for computations, which solved the flow field by using the compressible Navier-Stokes (NS) equations in combination with the laminar turbulent model. 2.2. Computational domain and boundary conditions (BCs) The geometry of the micronozzle was based on the microthruster developed by Cen et al. [13] summarized in Tab. 1, whose experimental data were used as reference for the performance evaluation of the numerical code. With Table 1: Microthruster geometry [13].

Region

Characteristics

Micronozzle inlet cross section throat cross section exit cross section convergent angle, αconv divergent angle, αdiv

1120 µm × 120 µm 150 µm × 120 µm 1760 µm × 120 µm 45◦ 15◦

respect to the Cen’s micronozzle, the throat section has been characterized by means of a radius of curvature equal to 75 µm, while at the inlet eight equivalent microchannels of 90 µm width have been used in order to preserve the actual injection area, in combination with a mixing region of 180 µm length before the entrance into the convergent region. The computational domains used for 2D and 3D simulations, are shown in Fig. 1(a) and (b) respectively, which exploited the symmetry of the planar geometry. The 2D domain consisted of 23931 cells and extended 15Wexit down2D

(c)

3D (a)

(b)

Fig. 1: Computational domains: (a) 2D mesh; (b) 3D mesh, global view; (c) 3D mesh, focused view.

stream and 6.5Wexit upward, where Wexit is the width of the exit section. The 3D mesh was made of 516360 cells, and the outer domain extended 5Wexit upward, 10Wexit downstream and 50Hn sideways, where Hn is the nozzle depth. The

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definition of the refinement levels resulted from grid independence study (GIS) conducted based on the analysis of water vapor flow at m ˙ = 5 mg s−1 and pout = 500 Pa. The grid sensitivity was evaluated by comparing the boundary layer thicknesses at the nozzle exit. The grid independence study led to the choice of the meshes which provided the best trade-off between accuracy and computational cost. Concerning numerical schemes, the central upwind scheme of Kurganov and Tadmor [14] was used for the flux terms and the Total Variation Diminishing (TVD) van Leer limiter [15] for interpolation. Moreover, the Gauss linear scheme was used for both the divergence, the gradient and the Laplacian operators. Time derivatives were computed with the first order, bounded and implicit Euler scheme. The time step was determined based on a maximum Courant number Comax of 0.2. Simulations stopped when the steady solution was reached, at around t = 3 × 10−4 s. Concerning the boundary conditions, the continuum NS flow model was used in combination with a partial slip boundary condition at walls since the Knudsen number Kn into the divergent reached values in the range 1 × 10−3 − 1 × 10−1 . Water vapor was used as propellant, for which polynomial laws were used for the computation of the dynamic viscosity and the specific heat capacity at constant pressure, in combination with the Peng Robinson equation of state. The first order Maxwell slip model was used at walls with a tangential momentum accommodation coefficient (σT MAC ) set to 0.8. The value 0.8 was supposed to be the most physical one in reference of the slip regime of a generic gas flow on polished silicon microchannels. Despite this consideration, a sensitivity analysis was conducted at σT MAC = 0.5, 0.7 and 0.8. Results revealed a weak dependence of the solution on σT MAC in the investigated range, with a maximum variation of the thrust force less than 1%. The mass flow rate at the inlet m ˙ and the static pressure of pout = 500 Pa at the outlet were imposed as boundary conditions. A 1D model of steady-state boiling allowed for the Table 2: Test matrix of CFD simulations.

Test Case

Dimensions

Tinlet , [K]

˙ [kg/s] m,

poutlet , [Pa]

SIM1

2D

505.58

5 × 10−6

500

505.58

−6

500

−6

500

SIM2

2D

5 × 10

SIM3

3D

505.58

5 × 10

SIM4

3D

505.58

5 × 10−6

500

SIM5

3D

505.58

5 × 10−6

500

Slip Condition at sidewalls

Slip Condition at planar walls

Pure Slip Maxwell, σT MAC = 0.80 Pure Slip Maxwell, σT MAC = 0.80 Maxwell, σT MAC = 0.80

Pure Slip Pure Slip Maxwell, σT MAC = 0.80

estimation of the temperature of the superheated vapor entering the micronozzle, which was set at 505.58 K. Instead, the inlet mass flow rate was set at m ˙ = 5 × 10−6 kg s−1 , in accordance with the experimental data by Cen [13]. The test matrix of the CFD simulation is reported in Tab. 2, where Pure Slip refers the non-viscous boundary conditions. 3D simulations distinguished each other based on the slip conditions applied to the sidewalls and the parallel planar walls of the micronozzle (see Fig. 1(c)).

2.3. Performance estimation The thrust force F was estimated by means of a cell-based integration at the exit section, then the specific impulse was computed by definition as I sp = m˙ Fg0 , where g0  9.81 m s−2 is the gravitational acceleration at sea level. m ˙ , where m ˙ ideal was derived from the the Instead, the numerical discharge coefficient was defined as Cd = m˙ ideal chocked nozzle condition and the isentropic flow hypothesis, by using the the average total pressure at the inlet predicted by CFD simulations. Furthermore, the I sp -efficiency was predicted by comparing the thrust force resulting from the CFD computations, with the one obtained by setting the pure slip condition at walls at the same operating F where the subscript pureSlip denotes the non-viscous solution obtained by using the pure conditions, i.e. ηv = F pureS lip slip condition at walls. The boundary layer growth into the expander was evaluated by computing the the displacement

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and the momentum thicknesses δ∗ and θ at different axial stations, as follows:  ∞ ρU x  dy 1− δ∗ = (ρU x )∞ 0  ∞  U  ρU x  x θ= 1− dy U x,∞ 0 (ρU x )∞

(1) (2)

where the subscript x denotes the axial direction of the flow, and ∞ denotes the undisturbed flow condition. 3. Results and discussion The predicted nozzle performance were compared with the experimental findings by Cen et al. [13]. In [13], the thrust force Fexp was measured by means of an impinging plate placed into the plume at 2 mm away from the nozzle exit. Similarly, the total jet thrust F jet resulting from CFD computations was estimated by means of cell-based integration of the local jet thrust over section normal to the axial direction. Results are shown in Tab. 3, which points out that the 3D viscous solution (SIM5) provided the best agreement with experiments, while 2D computations (SIM1 and SIM2) overestimated the jet thrust with respect to the experimental thrust, as well as the 3D isentropic solution (SIM3) and the 3D solution with slip condition on the planar walls (SIM4). Table 3: CFD results and experimental data [13].

Test Case

Exp. Thrust [13], Fexp [mN]

CFD Jet Thrust, Fjet [mN]

CFD Thrust, F [mN]

Specific impulse, Isp [s]

Discharge coefficient, Cd [-]

Isp -efficiency, ηv [-]

SIM1 SIM2 SIM3 SIM4 SIM5

5.20 5.20 5.20 5.20 5.20

6.00 5.80 5.91 5.81 4.72

5.91 5.61 5.94 5.68 2.81

120.6 114.4 121.0 115.8 57.4

0.958 0.934 0.965 0.937 0.917

1 0.949 1 0.957 0.474

In the following two main aspects are highlighted: i) the different behavior of the supersonic flow in relation to the boundary conditions at walls; ii) the analysis of the performance variation of the planar micronozzle. The analysis started with the comparison between the solutions of the 2D computations, i.e. the test cases SIM1 and SIM2. The first one referred to the isentropic solution, since a pure slip condition was assumed at walls, while the second one concerned with the viscous solutions at walls. As evinced in Fig. 3, the maxwellian boundary condition at walls (SIM2) produced the establishment of a subsonic boundary layer which produced a reduction of the performance of about 5% highlighted in Tab. 3. The axial profile of the displacement and momentum thicknesses are respectively denoted by the blue and red dashed line with crosses of Fig. 2(b), where the zero represents the location of the nozzle throat. As expected, it resulted an almost linear growth of the subsonic boundary layer, which is highlighted by the contour map of the Mach number in Fig. 3(b) in comparison with the one in Fig. 3(a) referred to the isentropic flow (SIM1). Hence, 3D computations were performed in order to study the influence of the nozzle depth on the viscous losses and the performance of the thruster. At the beginning, 3D simulations were performed in ideal flow conditions (SIM3) and with pure slip on the planar walls (SIM4) to be compared with 2D computations, more specifically with SIM1 and SIM2 respectively. Results highlighted that the behavior of the flow inside the micronozzle is quite similar in both cases. This findings was confirmed by comparing the performances in Tab. 3, and the boundary layer thicknesses of Fig. 2(a) where both the red and the blue lines with crosses well overlap with the ones marked by diamonds. A different flow behavior between 2D and 3D simulations was retrieved in the plume outside the micronozzle. In fact, in both 2D test cases SIM1 and SIM2 the jet plume took the typical bell-shape contour (see Fig. 3, yet 3D simulations



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SIM5, width-wise SIM5, depth-wise SIM5, width-wise SIM5, depth-wise

0.5

0

(a)

,[m]

1

0.5

0

0.5

1

1.5 2 x-coord, [m]

2.5

3

0 3.5

10-3

SIM2 SIM5 SIM4 SIM2 SIM5 SIM4

1.5

1.5

1

*

*

1

10-4 2

10-4

2

1.5

,[m]

1.5

,[m]

10-4 2

10-4

2

1

0.5

0

(b)

,[m]

6

0.5

0

0.5

1

1.5 2 x-coord, [m]

2.5

3

0 3.5

10-3

Fig. 2: Axial profiles of the boundary layer thicknesses (BLTs) δ∗ and θ predicted by CFD computations: (a) comparison between the width-wise and depth-wise BLTs for SIM5. (b) comparison of the width-wise BLTs between all test cases.

(a)

(b) Fig. 3: Mach number contours of 2D simulations: (a) SIM1; (b) SIM2.

predicted the formation of a set of oblique shock waves as a consequence of the reflection of the expansion fan close to the lips of the micronozzle. In particular, in case of full slip condition (SIM3) the jet plume was stretched upward, as shown in Fig. 4. Consequently, the jet plume resulted more extended in the width-wise direction and restrained along the depth-wise direction with respect to the test case SIM4. On the contrary, when the maxwellian slip condition was

(a)

(b)

Fig. 4: 3D field of the Mach number of the test case SIM3.

applied only to the sidewalls of the micronozzle, the jet plume became more axisymmetric owing to the more intensive oblique shock wave system (see Fig. 5). Important insights about the influence of the small depth/long expander nozzle configuration were retrieved by means of 3D computations with maxwellian slip applied to the entire micronozzle (SIM5). Fig. 6 underlines the

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(a)

(b)

Fig. 5: 3D field of the Mach number of the test case SIM4.

establishment of the nozzle blockage phenomenon, which produced a strong decrease of the Mach number during the expansion into the divergent section and the rapid growth of the boundary layer thicknesses along the widthwise direction, as reported in Fig. 2(a) by the red and blue lines denoted by circles. In particular, the growth of the thicknesses along the micronozzle exhibited a peak before the nozzle exit corresponding the maximum thickness of the subsonic pocket showed in Fig. 6(b)-Plane A. Instead, along the depth-wise direction the displacement thickness

(a)

(b)

Fig. 6: 3D field of the Mach number of the test case SIM5.

δ∗ reached about 20 µm, which corresponds to almost one third of the semi-depth of the micronozzle. The nozzle blockage also caused the viscous heating highlighted in Fig. 7 which contributed to the thermal chocking of the supersonic flow, and consequently, to the performance losses pointed out in Tab. 3. In fact, the thrust force and

(a)

(b)

Fig. 7: Contours of the temperature field on the symmetry plane A: (a) test case SIM4; (b) test case SIM5.

the specific impulse reduced of about the 47%, which corresponds to the I sp efficiency ηv , as well as the discharge coefficient Cd highlighted that mass flow rate reduced with respect to the ideal one of about the 92%. Also, by analyzing the jet plume of Fig. 6 another peculiar phenomenon was predicted. In fact, due to the stronger expansion ratio at the nozzle exit, the shock wave system gained enough energy to caused the formation of a small Mach disk formed inside the plume followed by a subsonic recirculation region. Concerning the shape of the plume, it was stretched similarly to SIM4, even though in the depth-wise direction probably due to the complex flow conditions induced by the Mach disk.

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4. Conclusions The present work provided a numerical investigation of a supersonic flow of a rarefied gas inside a planar micronozzle in slip regime. It focused on the influence of the partial slip condition at walls on the viscous losses and the boundary layer growth, in both 2D and 3D configurations. In particular, owing to establishment of the slip regime, the continuum Navier-Stokes equations were solved in combination with the Maxwell slip model at walls, setup with a tangential momentum accommodation coefficient equal to 0.8. The analysis was extended by simulating the supersonic flow in pure slip conditions, which allowed for the estimation of the global performance of the micronozzle in terms of thrust force, specific impulse, discharge coefficient and I sp -efficiency. Results showed that by neglecting nozzle depth, 3D simulations led to the same solution obtained by means of 2D computations inside the micronozzle. In particular, a linear growth of the boundary layer thicknesses on the sidewalls was predicted, and the viscous losses produced a reduction of the performance of about the 95%. Instead, significant differences were found in the prediction of the jet plume, which took the typical bell-shape form in cases involving 2D computations, yet 3D simulations estimated a plume characterized by the succession of oblique shock waves and expansion fan waves. The influence of the nozzle depth was considered by means of 3D computations with partial slip conditions applied to all the inner walls of the micronozzle. Results pointed out the establishment of the nozzle blockage and a viscous heating, which heavily affected the boundary layer growth on both the width-wise and the depth-wise directions, as well as the micronozzle performance due to a viscous-thermal chocking. In particular, the boundary layer thicknesses rapidly growth into the expander up to reach a maximum and then slightly decreased. This corresponded to the development of a hot subsonic pocket on sidewalls which modified the actual geometry of the micronozzle. The performance decreased of about the 47%, and the analysis of the jet plume highlighted the formation of the Mach disk followed by the typical diamond-shaped subsonic recirculation region. References [1] M. Silva, D. Guerrieri, A. Cervone, E. Gill, A review of mems micropropulsion technologies for cubesats and pocketqubes, Acta Astronautica 143 (2018) 234–243. doi:10.1016/j.actaastro.2017.11.049. [2] R. L. Bayt, K. S. Breuer, Viscous effects in supersonic mems-fabricated micronozzles, Vol. 66, 1998, pp. 117–123. [3] W. Louisos, D. Hitt, Viscous effects on performance of two-dimensional supersonic linear micronozzles, Journal of Spacecraft and Rockets 45 (4) (2008) 706–715. doi:10.2514/1.33434. [4] W. Louisos, D. Hitt, Performance characteristics of 3d supersonic micronozzles, 2008. [5] T. Handa, Y. Matsuda, Y. Egami, Phenomena peculiar to underexpanded flows in supersonic micronozzles, Microfluidics and Nanofluidics 20 (12). doi:10.1007/s10404-016-1831-1. [6] K. Cheah, J. Chin, Performance improvement on mems micropropulsion system through a novel two-depth micronozzle design, Acta Astronautica 69 (1-2) (2011) 59–70. doi:10.1016/j.actaastro.2011.02.018. [7] E. Sokolov, M. Chernyshov, Optimization of micronozzle performance at zero ambient pressure, Acta Astronauticadoi:10.1016/j. actaastro.2017.12.027. [8] G. Zhang, L. Wang, X. Zhang, M. Liu, Continuum-based model and its validity for micro-nozzle flows, Jisuan Wuli/Chinese Journal of Computational Physics 24 (5) (2007) 598–604. [9] G. A. Bird, Molecular gas dynamics, NASA STI/Recon Technical Report A 76. [10] H. Sofloo, R. Ebrahimi, A. Shams, Simulation of rarefied gas flows in mems/nems using a molecular method, Vol. 1, 2009, pp. 1039–1044. doi:10.1115/FEDSM2009-78015. [11] A. Alexeenko, D. Levin, S. Girnelshein’l, R. Collins, B. Reed, Numerical study of flow structure and thrust performance for 3-d mems-based nozzles, 2002. [12] M. Liu, X. Zhang, G. Zhang, Y. Chen, Study on micronozzle flow and propulsion performance using dsmc and continuum methods, Acta Mechanica Sinica/Lixue Xuebao 22 (5) (2006) 409–416. doi:10.1007/s10409-006-0020-y. [13] J. Cen, J. Xu, Performance evaluation and flow visualization of a mems based vaporizing liquid micro-thruster, Acta Astronautica 67 (3-4) (2010) 468–482. doi:10.1016/j.actaastro.2010.04.009. [14] A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convectiondiffusion equations, Journal of Computational Physics 160 (1) (2000) 241 – 282. doi:https://doi.org/10.1006/jcph.2000.6459. [15] B. van Leer, Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov’s method, Journal of Computational Physics 32 (1) (1979) 101 – 136. doi:https://doi.org/10.1016/0021-9991(79)90145-1.